fulltext

Stochastic dynamics and the classical equations of motion

Department of Physics, University of Massachusetts at Lowell - Lowell, MA 01854, USA

(ricevuto il 29 Aprile 1966; approvato il 30 Ottobre 1996)

Summary. — Based on the principle of indistinguishability of identical particles the expression for the canonical distribution in gamma space is derived for a many-particle conservative system of known kinetic and potential energy. It is further demonstrated as a consequence of the statistical independence of the orthogonal components of particle momentum that the particle trajectories are described by Hamilton’s equations of motion. Consideration is given to the situation where the Hamiltonian is not only a function of the canonical variables but also of time.

PACS 02.50.Ey – Stochastic processes. PACS 05.20.Gg – Classical ensemble theory. PACS 05.30.Ch – Quantum ensemble theory.

1. – Introduction

In a previous paper [1] the interrelationship between mechanics and thermo-dynamics has been examined in order to demonstrate that the stochastic behavior of a many-particle system in statistical equilibrium is not in contradiction with equations of classical dynamics. In particular, it is demonstrated that as a consequence of the imposition of symmetry conditions for particle exchange, it is possible to derive the basic postulates of statistical mechanics, the microcanonical and canonical distributions.

This prior investigation has demonstrated the compatibility between the stochastical dynamical behavior of such a system and the formalism of the classical laws of motion. A deeper issue is concerned with whether it is possible to derive the basic classical equations at the microscopic level through the application of the ideas of symmetry. It is this question that the present paper addresses and it is shown, given certain restrictive conditions, that this is possible.

2. – The basic assumptions

The basic formulation of the laws of mechanics due to Newton involves the concept of force. Alternate formulations have been proposed based on the basic energy concept 727

rather than that of force. The best known of these is Hamilton’s variational principle involving the Lagrangian of the system. Mention is made of this technique because the initial basis of the present development also uses knowledge of the system kinetic and potential energies rather than the concept of force.

The system to be considered is composed of monatomic particles where the total energy H is a constant of the motion. The following basic concepts are used (vectors are designated in bold type):

a) The kinetic energy for a system of N particles is given by T 4

!

where prrepresents the momentum of the r-th particle, and q represents the 3N

particle coordinates in configuration space.

c) The state of the system is invariant for particle exchange.

As a consequence of these three premises it will be demonstrated that the probability density distribution for the system in gamma space will be canonical. This factor, taken in conjunction with the conservation of probability current in the space, leads to Hamilton’s equations.

3. – Construction of the probability density function

Consider the interaction of three particles: A, B, C. Particles A and B lie between the interaction range whereas A and C lie outside the interaction range. The motion of

A towards B in terms of classical mechanical concepts will result in the creation of

velocity correlations between the two particles, whereas correlation between A and C remains negligible. If B and C are interchanged, then consonant with condition c), the state of the system remains unchanged. However, the following situation results: the velocities of A and C are uncorrelated although the particles are now within the inter-action range, whereas those of A and B are correlated although they lie beyond the interaction range. Consequently, it must be concluded that the velocities of A, B, and C, and therefore for all the system particles, are uncorrelated. It also follows that the particle velocities are not correlated with their relative positions in configuration space. Consequently, the probability density r in gamma space is given by the relation

r 4

»

fr(pr) F(q) ,

(1)

where fr(pr) represents the probability density in 3-dimensional momentum space for

the r-th particle, and F(q) represents the probability density in 3N configuration space for the 3N particle components qrs, r 41, R, N, s41, 2, 3.

In addition, the total system energy H is equal to the sum of the particle kinetic energies plus the potential energy of the whole system,

!

!

rsO2 m 1 V(q) 4 H ,

(2)

where prs(s 41, 2, 3) represents the three momentum components of the r-th particle

that H is a constant of the motion, is related to the p, q coordinates by the following equations: ¯fr(pr) ¯prs 1 lfrprsOm 4 0 , r 41, R, N , s41, 2, 3 , (3) ¯F(q) ¯qrs 1 lF¯V(q) ¯qrs 4 0 , r 41, R, N , s41, 2, 3 . (4)

Consider the integration of eq. (3) for r 41,

f1(p1) 4g1(p12, p13) exp [2lp112O2 m] , (5) f1(p1) 4g2(p13, p11) exp [2lp122O2 m] , (6) f1(p1) 4g3(p11, p12) exp [2lp132O2 m] , (7)

where g1, g2, g3represent functions such that each term on the left-hand sides of (5), (6)

and (7) are equal to one another. Considering eq. (5), it is evident that p11 must be

statistically independent of p12, p13. Similarly, from (6) and (7) p12is independent of p11, p13and p13is independent of p11, p12. Thus p11, p12, p13are statistically independent of

one another and accordingly f1(p1) must be equal to the product of the three

distribution functions for the momentum components, i.e.

f1(p1) 4A1exp [2l(p112O2 m 1 p122O2 m 1 p132O2 m) ] .

(8)

It will be noted from eqs. (5) and (7) that the principle of invariance under exchange of particles also leads to the statistical independence of the 3N orthogonal components of the particle momenta, a crucial factor in the determination of the equations of motion given below. Furthermore, from eq. (4) it follows that

F(q) 4B exp [2lV(q) ] ,

(9)

where B is a constant. From eqs. (1), (2), (8) and (9),

r 4C exp [2lH] .

(10)

The constants C and l represent the thermodynamic analogues exp [COkT ] and 1OkT, respectively, where C is the Helmholtz function, k is Boltzmann’s constant and T is the absolute temperature. Hence

r 4exp [ (C2H)OkT] .

(11)

4. – Derivation of the equations of motion

In addition to the expression for r, the ensemble probability density, one must add the condition that the members of the ensemble described by r are neither created nor destroyed, i.e.

r._{1˜(r j}._{) 40 ,}

where ˜ represents the 6N-dimensional divergence operator and j represents the 6N coordinates and momenta. Translated into Cartesian coordinates, eq. (12) becomes

r.₁

!

!

g

h

!

!

u

v

In equilibrium r._{4 0 . Furthermore,} q.rs4 prsOm in Cartesian coordinates, ¯r ¯prs 4 2 rprsOmkT , and ¯r ¯qrs 4 2 r ¯H ¯qrs

OkT . Hence eq. (13) becomes

2

!

!

g

h

N

!

!

u

v

Since q.rs4 prsOm , then the terms ¯q.rs

¯qrs

in the second summation are zero. Also, provided no velocity-dependent forces are present, then p.rs is a function of the q coordinates only, and the terms ¯p

.

rs

¯prs

are also zero. Furthermore, each bracket

(

¯qrs 1 p

.

rs

)

Conse-quently, since the canonical variables p, q are independent, and the momentum components prs are also statistically independent of one another, each bracket

(

¯qrs

1 p.rs

)

p.rs4 2 ¯H ¯qrs , q.rs4 prsOm 4 ¯H ¯prs . (15) 5. – Conclusion

Fundamental to the development given above is the demonstration that the orthogonal components of the particle momenta are all statistically independent. This constitutes a much stronger statement than the original premise involving the statistical independence of the particle momenta. It is interesting to note that historically Maxwell first derived the molecular velocity distribution for a gas based on the independence of the three velocity components of an individual particle. At the time this derivation was not considered to be sufficiently rigorous and was superseded by a stronger derivation based upon the principle of molecular chaos.

The microscopic equations of motion have been derived partly as a consequence of the properties of a conservative macroscopic system in a state of equilibrium. However, similar reasoning could have applied in the case of a slowly evolving system not in equilibrium but where the conditions for local equilibrium apply [1]. To treat the problem, the system under consideration is divided into cells small enough so that the thermodynamic properties vary little over each cell but large enough so that the cells can be considered as macroscopic subsystems. Each cell can then be effectively

considered as being in equilibrium except that the temperature and potential energy V (and therefore the total energy H) would be slowly varying functions of time. The resultant equations of motion would then be of the more general Hamiltonian form in which H is a function of p, q, t.

* * *

The author wishes to thank Dr. G. P. COUCHELLof the University of Massachusetts

Lowell for his patient and continuing interest, as well as his many valuable suggestions.

R E F E R E N C E S