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XXXIII
Dissipation Function Theory: a
mathematical foundation of Non
Equilibrium Statistical Physics and
Response Theory
Author:
Salvatore C
ARUSOSupervisor:
Prof. Claudio G
IBERTIProf. Lamberto R
ONDONIA thesis submitted in fulfillment of the requirements
for the degree of Dottorato di Ricerca in Matematica
COORDINATORE: Prof. Cristian Giardinà
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Declaration of Authorship
I, Salvatore CARUSO, declare that this thesis titled, “Dissipation Function Theory: a math-ematical foundation of Non Equilibrium Statistical Physics and Response Theory” and the work presented in it are my own. I confirm that:
• This work was done wholly or mainly while in candidature for a research degree at this University.
• Where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated. • Where I have consulted the published work of others, this is always clearly attributed. • Where I have quoted from the work of others, the source is always given. With the
exception of such quotations, this thesis is entirely my own work. • I have acknowledged all main sources of help.
• Where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself.
Signed: Date:
v
“We’ve got no money, so we’ve got to think.”
Ernest Rutherford
“What is the difference between theoretical physics and mathematical physics? Answer: Theoretical physics is done by physicists who lack the necessary skills to do real experiments; mathematical physics is done by mathematicians who lack the necessary skills to do real mathematics.”
vii
Università degli Studi di Modena e Reggio Emilia
Abstract
Dipartimento di scienze fisiche, informatiche e matematiche
Dottorato di Ricerca in Matematica COORDINATORE: Prof. Cristian Giardinà
Dissipation Function Theory: a mathematical foundation of Non Equilibrium Statistical Physics and Response Theory
The foundations of the Laws of Thermodynamics stand in center of the intellectual building of Physics, since their early formulation in late XIX century. Because of their central rele-vance, these concepts still spark flaming debates and propel profound discussions. Given that this is true in the realm of Equilibrium Statistical Physics – an established corpus of co-herent concepts – the situation is even more volatile in Non Equilibrium Statistical Physics, where foundational debates are still in progress. This state of things motivates the study of the theoretical and mathematical foundations of Non Equilibrium Statistical Physics. Besides the fascinating scientific aspects, such studies are made necessary by technolog-ical urgencies: bio- nano- technologies operate at a scale in which boundaries between macroscopic and microscopic are blurred, plus in these devices non equilibrium is the rule and not the exception. For all these reasons, we propose Dissipation Function Theory as a candidate base to lay the theoretical and mathematical foundations of Non Equilibrium Statistical Physics via a non perturbative response theory. In this work we start by dis-cussing Dissipation Function theory within the theoretical framework of Non Equilibrium Statistical Physics; we continue applying its formalism to Response Theory and numerical simulations of non equilibrium systems; in the final chapter we outline some possible fu-ture lines of research.
Sin dalle loro prime formulazioni nel tardo XIX secolo, le Leggi della Termodinamica sono centrali nell’impianto contettuale della Fisica. A causa della loro grande rilevanza, questi concetti sono tuttora causa di accesi dibattiti. Considerato che questo è vero per la Fisica Statistica di Equilibrio – un corpus coerente di concetti e metodi – la situazione è persino più volatile nel caso della Fisica Statistica di Non Equilibrio, dove i dibattiti fondazion-ali sono tuttora in corso. Questo stato di cose motiva lo studio delle fondamenta teoriche e matematiche della Fisica Statistica di Non Equilibrio. Oltre agli aspetti squisitamente scien-tifici, tali studi sono resi necessari dalle applicazioni tecnologiche: le bio- nano-tecnologie operano ad una scala alla quale la distinzione tra macroscopico è microscopico non è così netta, per di più in queste applicazioni il non equilibrio è la regola e non l’eccezione. Per tutte queste ragioni, proponiamo la Teoria della Funzione di Dissipazione come la teoria candidata per porre le basi della Fisica Statistica di Non Equilibrio utilizzando una teo-ria della risposta non-perturbativa. In questo lavoro di tesi iniziamo discutendo la Teoteo-ria della Funzione di Dissipazione inquadrandola nella cornice teorica della Fisica Statistica; continuiamo applicandone il formalismo alla teoria della risposta e a delle simulazioni nu-meriche di sistemi fuori dall’equilibrio termodinamico; nel capitolo finale indichiamo pos-sibili future linee di ricerca.
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Acknowledgements
I primi ringraziamenti vanno ai miei supervisor, prof. C. Giberti e prof. L. Rondoni: per la loro pazienza e per la loro guida. Entrambi sanno quanto sia culturalmente e scientifica-mente in debito nei loro confronti.
Ringrazio poi i miei genitori, Sebastiano e Lucia, senza i quali sarebbe stato impossibile arrivare fin qui.
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Contents
Declaration of Authorship iii
Abstract viii
Acknowledgements ix
1 From the II Law of Thermodynamics to Fluctuations Relations 1
1.1 Introduction . . . 1
1.2 The Dissipation Function. . . 3
Observables . . . 7
Irreversibility and Fluctuations Relations . . . 9
T-Mixing and ΩT-Mixing . . . 12
1.3 Entropy production in Statistical Physics . . . 15
1.4 Modelling a system along with its environment. . . 19
1.4.1 Gauss Least Constraint principle . . . 20
1.4.2 Nosé-Hoover thermostat . . . 22
2 Response Theory 25 2.1 Introduction . . . 25
2.2 Heisemberg and Schröedinger pictures . . . 26
2.3 Green-Kubo approach . . . 28
2.4 Exact Response and Green-Kubo Formalism . . . 35
2.5 Conclusions . . . 38
3 Numerical Simulations 41 3.1 Introduction . . . 41
3.2 Oscillator in a heath bath . . . 42
3.2.1 Equipartition Theorem . . . 45
3.2.2 Numerical Simulations description . . . 48
3.3 The FPUT chain . . . 59
3.3.1 Numerical simulations description . . . 62
3.3.2 Exploratory data analysis . . . 64
Phase Space and probability density function of phase space coordi-nates . . . 64
Pearson R coefficient . . . 65
Time lagged cross correlation . . . 66
3.3.3 Baseline temperature T = 100 - Difference Temperature ∆T = 10 . . . 67
3.3.4 Baseline temperature T = 100 - Difference Temperature ∆T = 50 . . . 78
3.3.5 Baseline temperature T = 100 - Difference Temperature ∆T = 90 . . . 90
4 Theoretical applications 103
4.1 Introduction . . . 103
4.2 Formal Thermodynamics . . . 104
4.2.1 The Case with Constant or Slowly-Varying Volume Contraction . . . . 107
Example 1 . . . 109
Example 2 . . . 109
4.3 Simple Attractors and Small Dissipation . . . 111
4.3.1 Simple Attractors . . . 111
Remark. . . 112
4.3.2 Small Dissipation . . . 113
4.4 Relationship between dynamics and probability density . . . 114
4.5 Dissipation Function, Wave Turbulence and β-FPUT . . . 122
4.6 Ensembles and Dissipation Function . . . 127
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List of Figures
3.1 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1and T0 = 100. On top, we have the time series
respec-tively of the kinetic energy virial and the potential energy virial. As we can see, these both settle to the equilibrium temperature value imposed by the external heath bath. In the inferior panel, we have the instantaneous value of dissipation and its time average: since the system is in thermodynamic equi-librium with the heat bath, this is zero. Time is measured in: n ∗ h, where n is the number of steps and h = 10−3is the length of a time step. . . 50
3.2 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1 and T0 = 100. On the left, we have the
probabil-ity densprobabil-ity functions of the momentum and coordinate respectively. On the right there is the phase portrait of the oscillator. In this way we obtain re-spectively: i. the marginal densities of the full probability density function in Phase Space, ii. a two dimensional slice of the full attractor in the three dimensional Phase Space M. . . 51
3.3 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1 and T0 = 100. On the top half, from the left to the
right, we have the probability density function of the instantaneous Dissipa-tion FuncDissipa-tion Ωf0 in different time instants. Being the system in equilibrium, the probability density function converges towards being centered in zero. The fact that the probability density function in the final time instant is cen-tered in zero and shows some variance. Considered the scale of the vertical axis, this fact can be probably attributed to numerical noise. On the bottom half, from the left to the right, we have the probability density function of the time average dissipation (1/t)R0
−tΩ
f0(Sτx)dτ in different time instants. For sake of readability we have indicated the same quantity with the symbol Σ0,t. We see that the support of these probability density functions is
posi-tive. Furthermore, we observe that the more time progresses the more the probability density functions are peaked on zero, being the system in equi-librium with the heat bath. The fact that the probability density function in the final time instant is strongly peaked in zero and shows some variance on its support towards positive values, can be attributed to the fluctuations induced by the heat contact with the bath. . . 52
3.4 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1and T0 = 100, and subject to a time dependent forcing
in the form F (t) = A cos(10t), with A = 20. Note that in this case the ratio A/(kBT0)2 is equal to A/(kBT0)2 = 0.002. On top, we have the time series
re-spectively of the kinetic energy virial hp∂pHiand the potential energy virial
hq∂qHi. We can clearly see that in this case the that equipartition theorem
does not hold: the potential energy virial hq∂qHi settles to a value which is
different from the equilibrium temperature value. This discrepancy can be understood in terms of the response integral showed above. In the inferior panel, we have the instantaneous value of dissipation and its time average: since the system is not in thermodynamic equilibrium both these quantities differ from zero. Note that we have multiplied the time average of the in-tegrated Dissipation Function by kBT0, since it is this quantity that has an
important physical significance: the time average of the integrated Dissipa-tion FuncDissipa-tion multiplied by kBT measures the dissipated energy. The last
plot regards the sum of the potential energy virial with the e time average of the integrated Dissipation Function by kBT0: interestingly this quantities
sum to the equilibrium temperature value and oscillate around it. Recall that T0 = 100is the heat bath temperature. Time is measured in: n ∗ h, where n is
the number of steps and h = 10−3is the length of a time step. . . . 53
3.5 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1and T0 = 100, and subject to a time dependent forcing
in the form F (t) = A cos(10t), with A = 20. Note that in this case the ratio A/(kBT0)2 is equal to A/(kBT0)2 = 0.002. On the left, we have the
proba-bility density functions of the momentum and coordinate respectively. On the right there is the phase portrait of the oscillator. In this way we obtain respectively: i. the marginal densities of the full probability density function in Phase Space, ii. a two dimensional slice of the full attractor in the five dimensional Phase Space E. . . 54
xv
3.6 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1and T0 = 100, and subject to a time dependent forcing
in the form F (t) = A cos(10t), with A = 20. Note that in this case the ratio A/(kBT0)2is equal to A/(kBT0)2 = 0.002. On the top half, from the left to the
right, we have the probability density function of the instantaneous dissipa-tion Ωf0 in different time instants. Since the system is out of equilibrium, the probability density function converges towards being centered and strongly peaked on a positive value. The fact that the probability density function in the final time instant is centered in a positive value and shows some vari-ance, considered the scale of the vertical axis, can be attributed to numerical noise. On the bottom half, from the left to the right, we have the probabil-ity densprobabil-ity function of the time average dissipation (1/t)R0
−tΩ
f0(Sτx)dτ in different time instants. For sake of readability we have indicated the same quantity with the symbol Σ0,t. We see that the support of these probability
density functions is positive. Furthermore, we observe that the more time progresses the more the probability density functions are peaked on a pos-itive value different from zero, being the system out of equilibrium subject to an external forcing and releasing heat towards the heat bath. The fact that the probability density function in the final time instant is strongly peaked in a value different from zero and shows some variance on its support to-wards positive values, can be attributed to the fluctuations induced by the heat contact with the bath. . . 55
3.7 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1and T0 = 100, and subject to a time dependent forcing
in the form F (t) = A cos(10t), with A = 40. Note that in this case the ratio A/(kBT0)2is equal to A/(kBT0)2 = 0.004. On top, we have the time series
re-spectively of the kinetic energy virial hp∂pHiand the potential energy virial
hq∂qHi. We can clearly see that in this case the that equipartition theorem
does not hold: the potential energy virial hq∂qHisettles to a value which is
different from the equilibrium temperature value. This discrepancy can be understood in terms of the response integral showed above. In the inferior panel, we have the instantaneous value of dissipation and its time average: since the system is not in thermodynamic equilibrium both these quantities differ from zero. Note that we have multiplied the time average of the in-tegrated Dissipation Function by kBT0, since it is this quantity that has an
important physical significance: the time average of the integrated Dissipa-tion FuncDissipa-tion multiplied by kBT measures the dissipated energy. The last
plot regards the sum of the potential energy virial with the e time average of the integrated Dissipation Function by kBT0: interestingly this quantities do
not sum to the equilibrium temperature value and oscillate around it. Time is measured in: n ∗ h, where n is the number of steps and h = 10−3 is the length of a time step. . . 56
3.8 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1and T0 = 100, and subject to a time dependent forcing
in the form F (t) = A cos(10t), with A = 40. Note that in this case the ratio A/(kBT0)2 is equal to A/(kBT0)2 = 0.004. On the left, we have the
proba-bility density functions of the momentum and coordinate respectively. On the right there is the phase portrait of the oscillator. In this way we obtain respectively: i. the marginal densities of the full probability density function in Phase Space, ii. a two dimensional slice of the full attractor in the five dimensional Phase Space E. . . 57
3.9 These quantities regard an anharmonic oscillator in thermal contact with a heat bath with kB = 1 and T0 = 100, and subject to a time dependent
forc-ing in the form F (t) = A cos(10t), with A = 40. Note that in this case the ratio A/(kBT0)2 is equal to A/(kBT0)2 = 0.004. On the top half, from the
left to the right, we have the probability density function of the instanta-neous dissipation Ωf0 in different time instants. Since the system is out of equilibrium, the probability density function converges towards being cen-tered and strongly peaked on a positive value. The fact that the probability density function in the final time instant is centered in a positive value and shows some variance, considered the scale of the vertical axis, can be prob-ably attributed to numerical noise. On the bottom half, from the left to the right, we have the probability density function of the time average dissipa-tion (1/t)R−t0 Ωf0(Sτx)dτ in different time instants. For sake of readability we have indicated the same quantity with the symbol Σ0,t. We see that the
support of these probability density functions is positive. Furthermore, we observe that the more time progresses the more the probability density func-tions are peaked on a positive value different from zero, being the system out of equilibrium subject to an external forcing and releasing heat towards the heat bath. The fact that the probability density function in the final time instant is strongly peaked in a value different from zero and shows some variance on its support towards positive values, can be attributed to the fluc-tuations induced by the heat contact with the bath.. . . 58
3.10 A representation of the FPUT-chain in physical space. The hot thermostatted particle – which models the hotter heat bath – is represented in red, whereas the cold thermostatted particle – which models the colder heat bath – is rep-resented in blue. The system itself is reprep-resented by the white coloured circles that go from the first particle to the N -th particle. The particles are obviously represented stationary in their equilibrium position. . . 64
3.11 Time series of the instantaneous Dissipation Function Ωf0 and of the time av-eraged Dissipation Function: 1/tRt
0 Ω
f0(Sτx)dτ , that for sake of readability we have indicated with Σ0,t. The system is subject to weak non equilibrium
conditions. For this reason the Dissipation Function oscillates slightly above zero.. . . 67
3.12 Projection of the overall attractor in Phase Space on to the 2-dimensional subspaces spanned by (q1, p1)through (q20, p20)directions. . . 68
3.13 Projection of the overall attractor in Phase Space on to the 2-dimensional subspaces spanned by (q21, p21)through (q40, p40)directions. . . 69
3.14 Probability density functions of the displacements in Phase Space. In this figure are reported the Probability density functions of q1through q20.. . . 69
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3.15 Probability density functions of the displacements in Phase Space. In this figure are reported the Probability density functions of q21through q40. . . 70
3.16 Probability density functions of the momenta in Phase Space. In this figure are reported the Probability density functions of p1through p20. . . 70
3.17 Probability density functions of the momenta in Phase Space. In this figure are reported the Probability density functions of p21through p40. . . 71
3.18 Pearson R correlation with a rolling window of 5∗104steps of the first particle
with the particles of the first half of the chain. Starting top left we have in the superior panel the two time series superimposed, in the lower the rolling windowed Pearson R correlation between: q1 − q5, q1 − q10. In the inferior
part we have the same quantities but for: q1− q15, q1 − q20. We see that high
levels of correlation are reached between q1− q5and q1− q10, especially in the
second half of the time evolution. The opposite can be said for q1 − q15 and
q1 − q20especially in the last segment of the time evolution. . . 71
3.19 Pearson R correlation with a rolling window of 5∗104steps of the first particle
with the particles of the first half of the chain. Starting top left we have in the superior panel the two time series superimposed, in the lower the rolling windowed Pearson R correlation between: q1− q25, q1 − q30. In the inferior
part we have the same quantities but for: q1 − q35, q1 − q40. We see that in
these cases there is not a clear pattern for the correlation. Moreover these quantities oscillates strongly between positive and negative values. . . 72
3.20 Starting from the superior panel, from left to right we have the rolling win-dow time lagged cross correlation between q1− q5and q1− q10. In the bottom
panel from left to right we have the rolling window time lagged cross corre-lation between q1− q15and q1− q20. This quantity is calculated starting from
time t = 6 ∗ 105, with window size 104 and time step t = 5 ∗ 103. These plots
are interesting as they show a clear pattern: starting from the first two pan-els the signal that is in second place leaves room as the driving to the first signal, this being represented by red bands disappearing in favour of blue bands. We see that this phenomenon is particularly visible at the extreme temporal segments of the offset. On the x-axis we have the offset between the two signals; the window width is chosen to be w = 104 time steps long.
On the y-axis – dubbed as epochs – we have the time chunks the whole time duration is splitted into as consequence of dividing it into equally sized time windows. The first epoch starts at time ts = 6 ∗ 105 and finishes at ts+ wans
3.21 Starting from the superior panel, from left to right we have the rolling win-dow time lagged cross correlation between q1 and q25, q1 and q30. In the
bot-tom panel from left to right we have the rolling window time lagged cross correlation between q1and q35, q1and q40. This quantity is calculated starting
from time t = 6 ∗ 105, with window size 104 and time step t = 5 ∗ 103. In the
two superior panels we can see how the role of the driving-responding par-ticle basically shifts, this is particularly evident examining the center of the offset bands. In the two inferior panels the is not a dynamics as clear as the one seen in the superior panels; this may be due to the fact that q1and q35, q1
and q40are well distanced on the chain. On the x-axis we have the offset
be-tween the two signals; the window width is chosen to be w = 104time steps
long. On the y-axis – dubbed as epochs – we have the time chunks the whole time duration is splitted into as consequence of dividing it into equally sized time windows. The first epoch starts at time ts = 6 ∗ 105and finishes at ts+ w
ans so on. . . 74
3.22 We have reported the time series of the virials of the thermostatted particles representative respectively of the hot heath bath and of the cold heath bath. We have fixed the kinetic temperature of the hot thermostat to Th = 110,
whereas the colder is fixed to Tc = 90. This can be easily seen from the
plots. We report such plots in order to verify that the dynamics we are sim-ulating actually respects the imposed boundaries conditions on kinetic tem-peratures. A feature worth noticing is that the kinetic energy virial and the potential energy virial strongly differ both in the hot and in the cold thermo-stat. We have seen that this is a signature of non equilibrium conditions. . . . 75
3.23 In this panel we have the kinetic and potential energy virials time series of the particles in the first half of the chain: from particle 1 to particle 20. It is interesting to note that the only particle that shows different values of the kinetic and potential energy virials is the particle 1. This particle is clearly out of thermodynamic equilibrium: it serves as an energy source, since it is linked to the hot heat bath. All the other particles show the kinetic and po-tential energy virials that after the initial transient, oscillate below the equi-librium, at around value T = 90. . . 76
3.24 In this panel we have the kinetic and potential energy virials time series of the particles in the second half of the chain: from particle 21 to particle 40. It is interesting to note that the only particle that shows different values of the kinetic and potential energy virials is the particle 40. This particle is out of thermodynamic equilibrium: this particle serves as an energy sink, since it is linked to the cold heat bath. All the other particles show the kinetic and potential energy virials that after the initial transient, oscillate below the equilibrium, at around value T = 90. . . 77
3.25 Time series of the instantaneous Dissipation Function Ωf0 and of the time av-eraged Dissipation Function: 1/tR0tΩf0(Sτx)dτ , that for sake of readability we have indicated with Σ0,t. The system is subject to mild non equilibrium
conditions. For this reason the Dissipation Function oscillates slightly above zero.. . . 78
3.26 Projection of the overall attractor in Phase Space on to the 2-dimensional subspaces spanned by (q1, p1)through (q20, p20)directions . . . 79
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3.27 Projection of the overall attractor in Phase Space on to the 2-dimensional subspaces spanned by (q21, p21)through (q40, p40)directions . . . 80
3.28 Probability density functions of the coordinates in Phase Space. In this figure are reported the Probability density functions of q1through q20. . . 80
3.29 Probability density functions of the coordinates in Phase Space. In this figure are reported the Probability density functions of q21through q40. . . 81
3.30 Probability density functions of the momenta in Phase Space. In this figure are reported the Probability density functions of p1through p20. . . 81
3.31 Probability density functions of the momenta in Phase Space. In this figure are reported the Probability density functions of p21through p40. . . 82
3.32 Pearson R correlation with a rolling window of 5∗104steps of the first particle
with the particles of the first half of the chain. Starting top left we have in the superior panel the two time series superimposed, in the lower the rolling windowed Pearson R correlation between: q1 − q5, q1 − q10. In the inferior
part we have the same quantities but for: q1 − q15, q1 − q20. We see that the
correlation between q1− q5is almost always positive, a part from some very
short segments in the fist part of the time evolution and in the last part. The correlation between q1 − q10 shows an oscillating pattern between positive
correlated phases and negative correlated ones. The correlations between q1 − q15and q1− q20show a more erratic behaviour. . . 83
3.33 Pearson R correlation with a rolling window of 5∗104steps of the first particle
with the particles of the first half of the chain. Starting top left we have in the superior panel the two time series superimposed, in the lower the rolling windowed Pearson R correlation between: q1− q35, q1 − q30. In the inferior
part we have the same quantities but for: q1−q35, q1−q40. In all these plots we
can see that the initial positive valued correlations are due to the similarity of initial conditions between the analyzed signals, afterwards it is very difficult to isolate a pattern. . . 84
3.34 Starting from the superior panel, from left to right we have the rolling win-dow time lagged cross correlation between q1− q5and q1− q10. In the bottom
panel from left to right we have the rolling window time lagged cross cor-relation between q1 − q15 and q1 − q20. This quantity is calculated starting
from time t = 6 ∗ 105, with window size 104 and time step t = 5 ∗ 103. In the
first plot we can see that the second signal is particularly dominant in the fist part of the time evolution, then we can see that it dynamically exchange the forcing-response role with the fist signal, especially in the final part of the time evolution and in the central segment of the offset window. The remain-ing plots do not offer any clear pattern as forcremain-ing-response roles change both with time and with the rolling offset. On the x-axis we have the offset be-tween the two signals; the window width is chosen to be w = 104 time steps
long. On the y-axis – dubbed as epochs – we have the time chunks the whole time duration is splitted into as consequence of dividing it into equally sized time windows. The first epoch starts at time ts = 6 ∗ 105 and finishes at ts+ w
3.35 Starting from the superior panel, from left to right we have the rolling win-dow time lagged cross correlation between q1−q25and q1−q30. In the bottom
panel from left to right we have the rolling window time lagged cross corre-lation between q1− q35and q1− q40. This quantity is calculated starting from
time t = 6 ∗ 105, with window size 104 and time step t = 5 ∗ 103. Also in
these cases it is really difficult to notice any notable pattern, with the only exception of the last panel where one can see a strong preponderance of the second signal in the first part of the dynamics and in the central segments of the offset: as already mentioned above, this may be due to proximity be-tween initial conditions of the two considered signals. On the x-axis we have the offset between the two signals; the window width is chosen to be w = 104
time steps long. On the y-axis – dubbed as epochs – we have the time chunks the whole time duration is splitted into as consequence of dividing it into equally sized time windows. The first epoch starts at time ts = 6 ∗ 105 and
finishes at ts+ wans so on. . . 86
3.36 We have reported the time series of the virials of the thermostatted particles representative respectively of the hot heath bath and of the cold heath bath. We have fixed the hot thermostat temperature to Th = 150, while the colder
is Tc = 50. From the plots, we can see that the dynamics we are
simulat-ing actually respects the imposed boundaries conditions on temperatures. A feature worth noticing is that the kinetic energy virial and the potential energy virial strongly differ both in the hot and in the cold thermostat. We have seen that this is a signature of non equilibrium conditions. . . 87
3.37 In this panel we have the kinetic and potential energy virials time series of the particles in the first half of the chain: from particle 1 to particle 20. It is interesting to note that the only particle that shows different values of the kinetic and potential energy virials is the particle 1. This particle is clearly out of thermodynamic equilibrium: this particle serves as an energy source, since it is linked to the hot heat bath. All the other particles show the kinetic and potential energy virials that after the initial transient, oscillate around an equilibrium value which is roughly equal to T = 80. Notably this equilib-rium value temperature is different from the initial temperature T0 used to
sample initial conditions from the extended canonical distribution. As it can be seen this temperature changes along the chain. . . 88
3.38 In this panel we have the kinetic and potential energy virials time series of the particles in the second half of the chain: from particle 21 to particle 40. It is interesting to note that the only particle that shows different values of the kinetic and potential energy virials is the particle 40. This particle is clearly out of thermodynamic equilibrium: this particle serves as an energy sink, since it is linked to the cold heat bath. All the other particles show the kinetic and potential energy virials that after the initial transient, oscillates around an equilibrium which is roughly equal to T = 80. . . 89
3.39 Time series of the instantaneous Dissipation Function Ωf0 and of the time av-eraged Dissipation Function: 1/tRt
0 Ω
f0(Sτx)dτ , that for sake of readability we have indicated with Σ0,t. The system is subject to strong non equilibrium
conditions. For this reason the Dissipation Function oscillates above zero. . . 90
3.40 Projection of the overall attractor in Phase Space on to the 2-dimensional subpaces spanned by (q1, p1)through (q20, p20)directions . . . 91
xxi
3.41 Projection of the overall attractor in Phase Space on to the 2-dimensional subpaces spanned by (q21, p21)through (q40, p40)directions . . . 92
3.42 Probability density functions of the coordinates in Phase Space. In this figure are reported the Probability density functions of q1through q20. . . 92
3.43 Probability density functions of the coordinates in Phase Space. In this figure are reported the Probability density functions of q21through q40. . . 93
3.44 Probability density functions of the momenta in Phase Space. In this figure are reported the Probability density functions of p1through p20. . . 93
3.45 Probability density functions of the momenta in Phase Space. In this figure are reported the Probability density functions of p21through p40. . . 94
3.46 Pearson R correlation with a rolling window of 5∗104steps of the first particle
with the particles of the first half of the chain. Starting top left we have in the superior panel the two time series superimposed, in the lower the rolling windowed Pearson R correlation between: q1 − q5, q1 − q10. In the
inferior part we have the same quantities but for: q1 − q15, q1 − q20. We see
that the correlation between q1 − q5 is almost always positive, a part from
some negative oscillations. The same can be said for the correlations between q1 − q10, q1 − q15and q1− q20especially in the last part of the time evolution. . 94
3.47 Pearson R correlation with a rolling window of 5∗104steps of the first particle
with the particles of the first half of the chain. Starting top left we have in the superior panel the two time series superimposed, in the lower the rolling windowed Pearson R correlation between: q1− q25, q1 − q30. In the inferior
part we have the same quantities but for: q1 − q35, q1− q40. As already seen
above, in these cases the correlations oscillate very strongly between positive and negative values. This may be due to the separation between the particles and the strong temperature gradient they are subject to. . . 95
3.48 Starting from the superior panel, from left to right we have the rolling win-dow time lagged cross correlation between q1− q5and q1− q10. In the bottom
panel from left to right we have the rolling window time lagged cross corre-lation between q1− q15and q1− q20. This quantity is calculated starting from
time t = 6 ∗ 105, with window size 104and time step t = 5 ∗ 103. We can sett
that there is a very strong interaction only in the case of q1− q5 with the
sec-ond signal being prevalent over the firs, as indicated by the prepsec-onderance of red coloured bands. On the other hand we can see that the strength of said interaction fades away proportionally with the distance between parti-cles. On the x-axis we have the offset between the two signals; the window width is chosen to be w = 104 time steps long. On the y-axis – dubbed as
epochs – we have the time chunks the whole time duration is splitted into as consequence of dividing it into equally sized time windows. The first epoch starts at time ts= 6 ∗ 105 and finishes at ts+ wans so on. . . 96
3.49 Starting from the superior panel, from left to right we have the rolling win-dow time lagged cross correlation between q1−q25and q1−q30. In the bottom
panel from left to right we have the rolling window time lagged cross cor-relation between q1 − q35 and q1 − q40. This quantity is calculated starting
from time t = 6 ∗ 105, with window size 104 and time step t = 5 ∗ 103. In all
these panels we can confirm what said in the previous one and in particular that strength of the interaction fades away proportionally with the distance between particles. This may be attributed to the strong temperature gradient the chain is subject to, that plays a major role in the dynamics of the chain. On the x-axis we have the offset between the two signals; the window width is chosen to be w = 104 time steps long. On the y-axis – dubbed as epochs –
we have the time chunks the whole time duration is splitted into as conse-quence of dividing it into equally sized time windows. The first epoch starts at time ts= 6 ∗ 105 and finishes at ts+ wans so on. . . 97
3.50 We have reported the time series of the virials of the thermostatted particles representative respectively of the hot heath bath and of the cold heath bath. We have fixed the kinetic temperature of the hot thermostat to Th = 190,
whereas the colder is fixed to Tc= 10. The plots verify that the dynamics we
are simulating actually respects the imposed boundaries conditions on tem-peratures. A feature worth noticing is that the kinetic energy virial and the potential energy virial strongly differ both in the hot and in the cold thermo-stat. We have seen that this is a signature of non equilibrium conditions. In this case the chain is subject to unusually strong non equilibrium conditions. 98
3.51 In this panel we have the kinetic and potential energy virials time series of the particles in the first half of the chain: from particle 1 to particle 20. It is interesting to note that the only particle that shows different values of the kinetic and potential energy virials is the particle 1. This particle is clearly out of thermodynamic equilibrium: this particle serves as an energy source, since it is linked to the hot heat bath. All the other particles show the kinetic and potential energy virials that after the initial transient, oscillate around an equilibrium value which increases gradually along the chain. Notably this equilibrium value temperature is different from the initial temperature T0 used to sample initial conditions from the extended canonical distribution. 99
3.52 In this panel we have the kinetic and potential energy virials time series of the particles in the second half of the chain: from particle 21 to particle 40. It is interesting to note that the only particle that shows different values of the kinetic and potential energy virials is the particle 40. This particle is clearly out of thermodynamic equilibrium: this particle serves as an energy sink, since it is linked to the cold heat bath. All the other particles show the kinetic and potential energy virials that after the initial transient, oscillate around an equilibrium value which increases gradually along the chain and reaches a maximum in the 38-th and 39-th particle. . . 100
xxiii
List of Abbreviations
FPU Fermi Pasta Ulam
1-d 1-dimesional
2-d 2-dimesional
RJ Rayleigh-Jeans
MEP Maximum Entropy Principle
GLE Generalized Liouville Equation NESS Non Equilibrium Steady State
MD Molecular Dynamics
NEMD Non Equilibrium Molecular Dynamics FR Fluctuations Relations
NPI Nonequilibrium Partition Identity TRI Time Reversal Invariant
xxv
1
Chapter 1
From the II Law of Thermodynamics to
Fluctuations Relations
1.1
Introduction
Since the 1980s, substantial advancements in the field of Non Equilibrium Statistical Me-chanics came from a combined effort. On one side there was a formal dynamical systems approach, on the other a computational effort, the latter being motivated by the increase of computing power and the ability to perform finer and larger Molecular Dynamics (MD) simulations. These studies consisted in an in depth analysis of Newton’s dynamical equa-tions for interacting particles systems, possibly subjected to external fields.
Obviously, these investigations result informative and practically useful, provided that quantum effects are irrelevant or negligible. For instance, the Fermi–Pasta–Ulam–Tsingou problem in this sense is a forerunner that paved the way to a tremendous number of math-ematical, computational and physical advances [40, 37]; see e.g. the books [35, 1, 90, 32,
4].
When interacting particles system is driven away from equilibrium by external forces that continually inject energy in it, in order for the system to avoid excessive heating, it has to be in contact with some external environment. This external environment continuously removes heat from it – only in such physical setting a nonequilibrium steady state may be reached. For this conceptual, yet very practical, reason one has to face the question regard-ing how to include in the description of the dynamics the very large – w.r.t. the system – environment [35]. One way to solve such this problem, consists in introducing synthetic forces in the equations of motion. These forces are meant to simulate the macroscopic ef-fect of a heat bath or a reservoir without simulating every atom of the environment. The implementation of these concepts via computer algorithms gave birth to the field of Non Equilibrium Molecular Dynamics (NEMD) as we know it[35,55]. In this setting a question rises spontaneously "how do we interpret such forces that do not exist in nature?". The answer consist in admitting that they are a “mathematical device used to transform a difficult boundary condition problem, the flow of heat in a system bounded by walls maintained at differ-ing temperatures, into a much simpler mechanical problem”, cf. the Introduction of Ref. [35]. In practice, these synthetic forces act constraining the dynamics of the particle systems at hand maintaining fixed some observable. In the following we will see two different, yet physically equivalent, methods: Gauss Least Constraint Principle and Nosé-Hoover ther-mostat. Such this modus operandi became so popular and widespread with time that the corresponding algorithms are now provided as default routines of MD packages [68,88].
The reason why NEMD became an extremely important framework can be easily un-derstood considering its effectiveness in computing transport coefficients in physically re-alistic models of fluids. Correspondingly, it was also made clear that the same physical observables could be computed with a formal approach resorting on dynamical systems theory [35]. At that point it was evident that the dynamical system approach to Non Equi-librium Statistical Mechanics based on Newton’s equations of motion was a viable route. In this context it was realized that the he opposite of the phase space contraction rate Λ, not only turns out to be proportional to the thermostatting forces, but may also be proportional to the entropy production, when local thermodynamics equilibrium applies, see [27]. For sake of completeness, we have to recall that a similar idea, had previously been considered by Andrej, who actually followed a different point of view [2]. In the work of this scholar it is possible to find the idea that the phase space contraction rate Λ could be related to the entropy time variation rate. This notion reached mainstream diffusion when Evans, Cohen and Morriss applied the representation of Sinai-Ruelle-Bowen measures to the fluctuations of the phase space contraction rate Λ in simulations of a shearing fluid subject to Gaussian isoenergetic thermostats [26],[84]. That paper is widely considered a landmark because of the fact that the authors derived and tested Fluctuation Relations (FR): a notion that stands among the few exact results of Non Equilibrium Statistical Physics. In particular it was established that, provided the dynamics of the model is time reversal invariant (TRI), the FR show that positive values of the energy dissipation are exponentially more likely than negative ones.
From the point of view of Thermodynamics, it was argued that FR constitute a possible explanation of the Second law of Thermodynamics. Such a strong indication led various authors to advance the idea of a perfect identification of the entropy production with the phase space contraction [41,78]. This was partially motivated by the appealing fact that the phase space contraction Λ is part of the standard machinery of dynamical systems theory, but it was later pointed out that its identification with entropy production is legitimate only in a limited number of situations [75,83]. Furthermore, time dependent settings and large fluctuations make the connection of Λ with physically relevant observables even weaker. And it was from NEMD that the evidence of the relevance of the Dissipation Function Ω fi-nally arrived. It was shown that this quantity preserves the meaning of energy dissipation rate even when Local Thermodynamic Equilibrium (LTE) is violated and thermodynamic quantities cannot be defined. Despite this important conceptual difference between the Dis-sipation Function Ω and the opposite of the Phase Space contraction rate −Λ, their steady state averages can be related to each other. Later it was finally recognized the importance of the Dissipation Function Ω.
The use of the Dissipation Function Ω provided also the mathematical foundations for a non-perturbative response theory. Furthermore, a physical interpretation of said Dissi-pation Function Ω allows to indicate that it has the same significance for non-equilibrium states thermodynamic potentials have in the equilibrium counterpart. The strength of Dis-sipation Function Ω resides in the fact that it can be used to formally solve the Liouville equation in terms of the flux operator of the dynamics and in terms of physical quantities, providing a bridge between the physics and the mathematical description of the model at hand. Consequently, it enables the study of the exact – i.e. non perturbative – average response of the observables of ensembles of systems. Arguably, an even more important feature of Dissipation Function Ω resides in the fact that – provided that some conditions are met – it can be used to provide conditions for the evolution of single systems, [54,36] as opposed to standard Response Theory that regards only ensembles of systems.
1.2. The Dissipation Function 3 Obviously all these theoretical features, come at a cost. In fact investigating the proper-ties of the Dissipation Function in realistic interecting particles models can become a very complicated task, for this reason one may turn its efforts to mathematically amenable dy-namical systems, even when their physical interpretation may not be immediate. These simplified models are not meant to explain any physical mechanism, on the contrary their utility consist in the fact that they help highlighting aspects of the relation between dynam-ics and thermodynamdynam-ics, an eminent example being Helmoltz’s theorem [44].
This chapter is organized as follows.
In section1.2 we introduce the main mathematical object of this thesis: the Dissipation Function. We explore its physical and mathematical significance starting from the general equation describing the evolution of a probability distribution function in phase space. From this, we give an account of the Dissipation Function in relation with phase variables and observables and we firstly introduce the Response Theory based on Dissipation Func-tion. Building from that, we face the problem of irreversibility from the point of view of Fluctuations Relations and finally introduce two notions that central in Response Theory based on Dissipation Function: T-mixing and ΩT-mixing.
In section1.3 the problem of entropy production in Statistical Physics is considered. In particular we adopt a microscopic perspective, that allows us to point out the pivotal role of Dissipation Function in those cases in which not even Local Thermodynamic Equilibrium is verified.
These paragraphs naturally bring us to the final section1.4, that is devoted to display-ing some of the mathematical modelldisplay-ing techniques available to describe a particle system when this is in thermal contact with a heat bath. In particular Gauss’ Least Constraint Principle and the Nosé-Hoover thermostatting mechanism are analyzed.
1.2
The Dissipation Function
Let us start in full generality and consider a system whose microstate in phase space M ⊂ Rn is given by the point x ∈ M. Let us suppose that this system evolves in time and its dynamics is described by the system of ODEs:
˙x = V(x); (1.1)
where V(x) is a well behaved vector field.
This dynamics will generate a flow on M, encoded by the map St: M → M, such that
given a point x, its time evolution is given by St(x), t ∈ R. This represents the solution to
the ODEs of (1.1) at time t with initial condition x.
In many cases of physical interest there is no simple solution available to these equa-tions.
In the remainder of this work, we will specify properties of the phase space M and of the vector field of the dynamics V(x) when needed.
We endow the phase space M with a probability measure µ0(x)absolutely continuous
w.r.t. Lebesgue measure dx. Given this probability measure, we explicit by f0(x)its density
function, in formula: dµ0(x) = f0(x)dx.
Recall that, a measure µ is called invariant if µ(S−tE) = µ(E)for any measurable set E ⊂ Mand for all t ∈ R.
If µ0 is not invariant it naturally evolves under the dynamics, so that at time t it is
expressed by µt(E) = µ0(S−tE). Provided that the flow is sufficiently smooth, µtis also
ab-solutely continuous, and has density ft(x)that satisfies the Generalized Liouville Equation
(GLE): ∂ft(x) ∂t = − (ft(x)∇ · V(x) + V(x) · ∇ft(x)) ft(x) t=0 = f0(x) (1.2)
Bearing this formula in mind, we are equipped with all the necessary elements to write the total time derivative of the probability density function:
dft(Stx) dt = ∂ft(Stx) ∂t + V(S tx) · ∇f t(Stx) = −Λ(Stx)ft(Stx) (1.3)
where Λ(x) denotes the divergence of the dynamics vector field, i.e. Λ(x) ≡ ∇ · V(x), and its negative −Λ(x) is known as the phase space expansion rate. The difference between eq. (1.2) and eq. (1.3) can be understood pointing out that that the former encodes the Eulerian time derivative, whereas the latter gives the Lagrangian time derivative.
This calculation informs us that the probability density ft(x) changes with time along
a trajectory whenever the divergence of the vector field Λ(x) is different from zero. Im-portantly, we have to avoid confusing geometrical and dynamical aspects implied in these considerations. First of all it is necessary to say that the condition Λ = 0 is ultimately a property of the coordinate system we adopt in describing the phase space, in relation with the explicit form of the vector field of the dynamics. Consider for instance, we are able to determine via a suitable diffeomorphism a coordinate system in which the invariant mea-sure is uniform w.r.t. a given dynamics – the case of canonically conjugate variables for Hamiltonian dynamics in the microcanonical ensemble being the most immediate exam-ple. In such case, the initial distribution will be transported unchanged along the flow. Nevertheless, this does not mean that a given probability density remains invariant with time, since the information carried by the equations above is local in nature and does not capture what does happen in other regions of phase space M. Specifically, trajectories whose initial conditions are located in regions of phase space M characterised by different densities, during their time evolution may get locally intertwined with each other, giving rise to distributions that strongly locally vary with time [54].
One is led to rewrite the GLE as follows: ∂ft
∂t (x) = ft(x)Ω
ft(x) (1.4)
where the Dissipation Function is defined as: Ωft(x) ≡ −[V · ∇ ln f
t(x) + Λ(x)]
In accordance with the GLE, the time evolution of an ensemble probability density func-tion can be formally represented via the use of the Dissipafunc-tion Funcfunc-tion, in particular via its time integral.
In more detail, from direct integration of eq. (1.3), it is possible to obtain the ensemble probability density function at time t > 0 at point Stx, provided that the trajectory had
1.2. The Dissipation Function 5 started from the generic initial point x at time t = 0, in formula:
ft(Stx) = f0(x) exp Z 0 −t Λ(Sτx)dx = f0(x) exp{Λ−t,0(x)} (1.5)
where we have introduced the shorthand notation for integrals calculated over Phase Space trajectories, namely if O : M → R is a phase function, its time integral over a phase space trajectory taking place in the time interval [0, t] is given by:
O0,t(x) =
Z t
0
O(Sτx)
dτ Equation (1.5) can be written equivalently as:
ft(Stx) = f0(x) exp{−Λ0,t(x)} (1.6)
where one has to note that the difference in the two equations – which are conceptually equivalent – consists in shifting the domain of integration of the phase space expansion rate time integral, compensating this fact with the multiplication by a minus sign. Despite this may seem a minor detail, we have to highlight it is not. In the expressions defining the ensemble probability density function at future times, Λ−t,0 is integrated forward in time
over trajectories that have their initial condition in the past. In formula: Z 0 −t Λ(Sτx)dτ = − Z t 0 Λ(Sτx)dτ
In order this step to be allowed, we have to assume the dynamics to be invertible, i.e. there is a unique projection of the dynamics backwards in time. Basically, this is equivalent to requiring that the flux operator itself belongs to a strongly continuous group of operators.
Now let us turn our attention to considering the case in which we are interested in finding the value of the time evolved ensemble probability density function at a single point x. In such this situation, we have to take into account the time integral of the Dissipation Function. Chosen a time T ∈ R, T > 0, the integrated Dissipation Function reads:
Ωft 0,T(x) = Z T 0 Ωft(Sτx)dτ = − Z T 0 [Λ(Sτx) + V(Sτx) · ∇ ln ft(Sτx)]dτ = = −Λ0,T(x) − Z T 0 V(Sτx) · ∇ ln ft(Sτx)dτ = −Λ0,T(x) − Z T 0 d dτ ln ft(S τx) dτ = = ln ft(x) ft(STx) − Λ0,T(x) (1.7) Note how the time integral of the Dissipation Function is constituted by the time integral of the phase space expansion rate plus a contribution that is given by the logarithm of the ratio of the probability density function calculated at the initial point x and in the final point STx. In this sense we have to consider the time integral of the Dissipation Function as an
object that depends not only on phase space points, but also on phase space trajectories. In order to take into account how a probability density function evolves in phase space,
let us now examine the action of the exponential of the integrated Dissipation Function on the time varying ensemble probability density function.
Given two real numbers r, t ∈ R, using eq. (1.7) and eq.(1.5) we can write: exp{Ωfr r,r+t(x)}fr(Sr+tx) = fr(Srx) fr(Sr+tx) exp{−Λr,r+t(x)} fr(Sr+tx) = = fr(Srx) exp{−Λr,r+t(x)} = fr+t(Sr+tx) (1.8)
Choosing the initial probability density function f0(x), we obtain a formal solution of
the GLE in terms of the time evolving probability density function remaining fixed on a point in phase space, explicitly:
ft(x) = f0(x) exp{Ω f0
−t,0(x)} (1.9)
This represents the result we obtain adopting the Eulerian approach in studying the time evolution of the probability density function. On the other hand, if we wanted to consider the time evolution of the probability density function whilst following a trajectory in Phase space – rephrasing: adopting a Lagrangian approach – the result would involve the time integral of the phase space expansion rate, as seen above.
We must point out an important detail here. In the same manner seen before, in order to obtain the expressions defining the ensemble probability density function at future times, we had to multiply the initial ensemble probability density function by the exponential on the time integral of Ωf0(x) and Λ(x) respectively. These integrals are computed forward
in time over trajectories whose initial condition is located in the past. As anticipated, this step is allowed, only if the dynamics is invertible, namely there is a unique projection of the dynamics backwards in time.
Given its close relationship with time evolving ensemble density functions, the Dissi-pation Function has a number of properties worth exploring. A very important relation is constituted by the non-equilibrium partition identity (NPI), which constitutes one of the first formal results regarding non-equilibrium Statistical Mechanics. Initially discovered in the framework of Hamiltonian systems [56],[95], it was later found to be an important expression obtainable from models of non equilibrium systems[64],[12]. This important result ensures that ensemble density functions evolving under non-equilibrium conditions are always normalized to one, i.e. RMft(x)dx = 1.
In particular, if the notation h.i0 indicates the phase average w.r.t the initial density, the
NPI reads: D exph−Ωf0 −t,0 iE 0 = 1 (1.10)
it interesting to note that this result is consistent with probability theory and in particular, with the notion of charateristic function.
Given a probability density function ft(x), its Fourier transform w.r.t. coordinates is
indicated as its charateristic function Φt(k):
Φt(k) = Z M ft(x)eık·xdx = Z M f0(x)eΩ f0 −t,0(x)+ık·xdx (1.11)
where the second equality is obtained expressing the time evolved probability density with the initial one via the exponential of the time integral of the Dissipation Function, i.e. eq. (1.9).
1.2. The Dissipation Function 7 From basic probability theory, know that that normalization of the probability density reflects on the charateristic function as follows:
1 = Φt(0) = Z M f0(x)eΩ f0 −t,0(x)dx = D exph−Ωf0 −t,0 iE 0 (1.12)
and this simple step directly yields the NPI from the normalization of the probability den-sity function.
Observables
In developing this section we have referred to [54].
As we learn from Equilibrium Statistical Mechanics, the relevance of a statistical ap-proach to dynamics resides in the fact that one can express physical observables as phase averages of suitably defined phase space functions.
For Equilibrium Statistical Mechanics the phase average of the phase function O : M → R is expressed as an expectation calculated using the equilibrium measure µeq. The
sub-script stresses the fact that we take into account the equilibrium stationary measure. Pro-vided that such this measure is continuous w.r.t the Lebesgue one, we can write:
hOi = Z
M
O(x)feq(x)dx (1.13)
In the case of Non Equilibrium Statistical Mechanics the temporal variable enters in the definition of phase averages. The phase average, calculated in the case of a nonequilibrium measure µtwhich is the temporal evolution of the initial measure µ0, will read:
hOit= Z
M
O(x)ft(x)dx (1.14)
Given that the time evolved density ftcan be expressed using the initial density f0and the
Dissipation Function, it is clear that the latter has a role in manipulating phase averages of observables and even their time derivatives.
Let us start this discussion introducing some notation that will be useful in manipulat-ing time dependent phase averages.
Given a phase function O : M → R, we denote its time integral as: O0,T(x) =
Z T
0
O(Sτx)
dτ
Using the group properties of the flux operator, we can rigidly shift the domain of integra-tion of the time integral by a finite time t1 > 0, counterbalancing this operation by a shift
backward in time of the flux operator: O0,T(x) = Z T 0 O(Sτx) dτ = Z T +t1 t1 O(Sτ −t1x)dτ = Z T +t1 t1 O(S−t1Sτx)dτ = = Ot1,T +t1(S −t1x) (1.15)
We start by decomposing the identity operator as the composition of the flux operator forward in time with its inverse backward in time – namely I = StS−t
= S−tSt – and performing the change of coordinates: y = S−rx, whose Jacobian determinant is
∂y ∂x = exp{Λ−r,0(x)}. We obtain: hOit+r = Z O(x)ft+r(x)dx = Z O(SrS−rx)ft+r(SrS−rx) ∂y ∂x dx (1.16)
By the use of eq.(1.15), we rewrite the Jacobian determinant as: Λ−r,0(x) = Λ0,r(S−rx) = Λ0,r(y)
And reinserting this in the the equation above, gives the expression of the phase average at time t, further advanced in time by the amount r:
hOit+r = Z
O(Sry)ft+r(Sry) exp{Λ0,r(y)}dy =
Z
O(Sry)ft(y)dy = hO ◦ Srit (1.17)
Note that in the second equality we have used eq. (1.5) in conjunction with (1.9).
This identity will now result useful in computing the time derivative of the time depen-dent phase average of an observable.
Consider in fact the explicit calculation of the time dependent phase average derivative: d hOir dr = limh→0 1 h Z O(x)[fr+h(x) − fr(x)]dx = = lim h→0 1 h Z
O(x)ft(x)[exp{Ωft−r−h,0t (x)} − exp{Ωft−r,0t (x)}]dx =
= lim
h→0
1 h
Z
O(x)ft(x) exp{Ωft−r,0t (x)}[exp{Ω ft
t−r−h,t−r(x)} − 1]dx
(1.18)
Where we observe that the second equality is justified using the representation of time evolved probability density functions enabled by Dissipation Function, see eq. (1.9).
So we now have to consider the limit under the sign of integral by itself. Obviously, this is a mathematically delicate situation and we suppose that all the conditions to perform the exchange in the order of performed operations are verified:
lim h→0 1 h[exp{Ω ft t−r−h,t−r(x)} − 1] = d dhexp{Ω ft t−r−h,t−r(x)} h=0 = = Ωft(St−r−hx) exp{Ωft t−r−h,t−r(x)} h=0 = Ωft(St−rx) (1.19)
Plugging this result back in the initial equation for the phase average time derivative of the observable, we have: d hOir dr = Z O(x)fr(x)Ωft(St−rx)dx =O(Ωft ◦ St−r) r (1.20)
1.2. The Dissipation Function 9 Now posing t = 0 and using eq.(1.15), we arrive to:
d hOir dr =O(Ω f0 ◦ S−r) r =(O ◦ S r)Ωf0 0
This is the pivotal equation that finally allows us to write the time evolving phase average of a phase observable as:
hOit= hOi0 + Z t
0
(O ◦ Sτ)Ωf0
0 dτ (1.21)
where the result is obtained by a direct integration in time. Irreversibility and Fluctuations Relations
Exaclty as the previous section, this one is largely based on [54].
We will treat Fluctuations Relations (FR) in the case of deterministic equations of mo-tion. FR and related theorems enable the use of Statistical Thermodynamics in non equi-librium experimental situations, where standard classical Thermodynamics cannot be em-ployed and large fluctuations are an important feature of the dynamics [32],[35],[91].
Results related to Dissipation Function are exact arbitrarily far from equilibrium where entropy, and its rate of production, cannot be defined and the use of Linear Irreversible Thermodynamics is not a viable option. This degree of effectiveness is achieved by consid-ering the use of phase space path integrals of phase functions, instead of the average of the same phase functions calculated using the stationary probability measures of Equilibrium Statistical Mechanics.
FR in their modern formulations revolve around two important notions: the Dissipation Function Ω and the time-reversal invariance of the dynamics. The concept of time-revesal invariance is better understood when considering mechanical N -particle systems, for this reason let us consider the following particularization of the dynamics in1.1:
˙qi = pi/m, i = 1, 2, ..., N ˙ pi = Fi (1.22)
Where Fi is the total force exerting its action on the i-th particle.
Given the notation, let us clarify the concept of time-reversed trajectories: 1. one starts considering the point in phase space
x = (q1, ..., qN, p1, ..., pN)
representative of the state of a N -particle system, given a dynamics in phase space; 2. its time evolved counterpart will be denoted via the flux operator, which acts on the
single phase variables via its projection. If we suppose that the dynamics unfolds in a 3-dimensional physical space, the projection operator Πi allows to denote the action
of the flux operator on a particular phase variable, namely Πi : M → R3. Hence, we
will write:
3. the time-reversed version of this quantity will be defined as the image of the time evolved phase point under the action of the time reversal operator I – i.e. IStx. In
the case of a N-particle system whose forces do not depend on velocities, the time reversal operator acts as indicated:
IStx = (q1◦ St, ..., qN ◦ St, −p1◦ St, ..., −pN ◦ St)
For the continuation of our discussion, the notion of time-revesal invariance (TRI) plays a pivotal role; let us suppose that a dynamical system is TRI, in practice this means that it satisfies the following operator identity:
IStx = S−tIx, ∀x ∈ M, t ∈ R (1.23)
Note that this notion regards the properties of the dynamics and the dynamics only. This distinction is crucial in the fact that, despite microscopic dynamics may exhibit TRI, macro-scopic observables may not. Hence the apparent paradox regarding how a system, whose underlying dynamics is supposedly TRI, can show irreversible behaviour. This question tormented the founders of Statistical Mechanics and FR are invested of such great impor-tance because they help in understanding and solving this very issue.
The approach implied by FR consists in acknowledging that dynamics is TRI, but in general we do not have access to the microstate of a given system. For this reason one shifts the focus on a set of physically relevant observables, and inspects the properties of these observables under the action of the of the time reversal operator I. The action of this operator naturally separates physically relevant observables into two distinct categories:
• Even observables. As the name suggest, these are observables that are symmetric, i.e. they remain unchanged, under the action of the time reversal operator, namely:
E(Ix) = E(x)
An example of this kind of observable being the squared modulus of a particle mo-mentum.
• Odd observables. On the other hand, these observables are anti-symmetric w.r.t. the action of the time reversal operator, namely:
O(Ix) = −O(x)
An example of this kind of observable being the particle momentum itself.
It is interesting to see that the Dissipation Function happens to be an odd observable. This, together with the fact that in many physically informative models we can assimilate the Dissipation Function with a generalization of the entropy production, is assumed to be an explanation of the II Law of Thermodynamics [32].
In order to continue this discussion, consider the following notation to indicate the open interval with center A ∈ R, A > 0 and radius δ ∈ R, δ > 0:
1.2. The Dissipation Function 11 Given an observable O : M → R, let us consider a set in M with the following defini-tion:
O
(A)δ = {x ∈ M : O(x) ∈ (A − δ, A + δ)}
The next step consist in considering the time average of the integrated Dissipation Func-tion Ωf0
0,T over the interval [0, T ]. This quantity will be denoted as:
¯ Ωf0 0,T(x) = 1 T Z T 0 Ωf0(Sτx)dτ
Now the remainder of this discussion will be devoted to studying the fluctuations of ¯Ωf0
0,T(x)
around an arbitrarily chosen positive value A ∈ R. This will prove useful in providing the formulation of the FR.
In order to obtain the modern form of the Transient Fluctuation theorem [54] we have to consider the ratio:
µ0 ¯Ωf0,T0 (A)δ µ0 ¯Ωf0,T0 (−A)δ = R ¯ Ωf00,T (A)δ f0(x)dx R ¯ Ωf00,T (−A)δ f0(x)dx (1.24) A closer examination of this formula informs us that it quantifies the ratio between the probability (w.r.t. the initial measure µ0) to observe a positive value of the time average of
the integrated Dissipation Function against the probability to observe a negative value of the same quantity.
Considered the symmetry properties of the Dissipation Function w.r.t. the action of the time reversal operator, one realizes that the two domain of integration are related by the coordinate transformation in the form:
y = ISTx (1.25)
in particular, via this transformation – whose Jacobian is eΛ0,T(x)– it is possible to map the
points of the set (−A)δin those of (A)δ.
The coordinate transformation has obviously a direct impact on the other quantities appearing in1.24leading to the following considerations.
The initial probability density function is assumed to be even w.r.t. the action of the time inversion operator:
f0(Ix) = f0(x)
This is a condition that is easily realized, considering for instance as initial distribution the Canonical ensemble.
On the other hand the time integral of Λ is odd: eΛ0,t(Ix) = e−Λ0,t(x)
Using the identities regarding the time evolved probability density functions, see eq.(1.9) and eq.(1.5), we have:
f0(Stx) = f0(x)e−Λ0,t(x)−Ω
f0 0,t(x)
