Hamiltonian case

In the Hamiltonian case we set u = XH− γ

µ 0

∇pH

¶

=

µ ∇pH

−∇qH − γ∇pH

¶

, (2.34)

where γ > 0 is a parameter. With the above vector field, from the equations ˙q = ∇pH, ˙p =

−∇qH − γ∇pH, it follows ˙H = −γ|∇pH|² ≤ 0. The dissipative force-term −γ∇pH is added in order to compensate, on average, the effect of the noise. For what concerns the latter, the rea-sonable choice consists in setting the q-component of the noiseξ^(q)≡ 0, so that only a stochastic force componentξ^(p) is present; we assumeD

ξ^(p)_i (s)E

= 0, D

ξ^(p)_i (s)ξ^(p)_j (s⁰)E

= 2Dδi jδ(s − s⁰). The latter choice is equivalent to take the noise correlation matrix in (2.33) of the form

G =

µ O_n O_n O_n 2D I_n

¶ .

Thus, we consider the “Hamiltonian” Langevin equations

½ q = ∇˙ pH

p = −∇˙ qH − γ∇pH + ξ^(p) , (2.35) whose corresponding Fokker-Planck equation reads

∂ρ

∂t = −{ρ, H} + ∇p· (γρ∇pH) + D∆pρ , (2.36) where∆p:= ∇p· ∇p=P_n

i=1∂²/∂p²_i. Equilibria

We now look for equilibria of the Fokker-Planck equation (2.36) in the form ρeq = f (H). By inserting the latter “ansatz” on the right hand side of (2.36) one gets

∇p·£(γf + D f⁰)∇pH¤ = (γf⁰+ D f⁰⁰)|∇pH|²+ (γ f + D f⁰)∆pH = 0 . (2.37) First of all, we notice that the above equation is identically satisfied if γf + D f⁰= 0. On the other hand, both the coefficients of |∇pH|² and of∆pH in the equation are functions of H. Let us assume that, for p close enough to zero, H admits the expansion H(q, p) = a(q) + b(q) · p +

1

2p · Q(q)p +O(|p|³), which is always true in the case of a smooth dependence of H on p; let us also define w(H) := γf (H) + D f⁰(H). Equation (2.37) reads

w⁰(H)£|b|²+O(|p|)¤ + w(H)[trQ +O(|p|)] = 0 (2.38) which holds iff w(H) = 0. Indeed, the sufficiency of the condition is obvious, while the necessity follows taking the limit p → 0 on the constant energy surface SE = {(q, p) : H = E}, where (2.37) yields

w⁰(E)|b|²+ w(E)trQ = 0 ,

2.4. APPROACH TO EQUILIBRIUM I: NOISE & DISSIPATION 33 which is generically impossible (the coefficients of the Hamiltonian should depend on the en-ergy level). Since w = γf + D f⁰, it follows that the Fokker-Planck equation admits a unique stationary solution of formρeq= f (H), namely the density

ρeq= 1

Z e^−DγH , (2.39)

if Z :=R

Γe^−DγHdV exists. In order to identify the density (2.39) with the canonical Gibbs one given in (2.16), it is necessary that the the diffusion coefficient D and the dissipation coefficient γ be not independent from each other, their ratio being the temperature of the system, namely

T =D

γ , (2.40)

which is known as the “fluctuation-dissipation relation”.

Free energy

We want to study the stability of the equilibrium solution (2.39). To such a purpose, we in-troduce a suitable Lyapunov function F(ρ) having a minimum at ρeq, and show that its time-derivative along the flow of the Fokker-Planck equation is nonpositive. If one defines the inter-nal energy U of the system as

U(ρ) :=

Z

ΓHρ dV , (2.41)

and the entropy S as

S(ρ) := − Z

Γρ logρ dV , (2.42)

the natural choice for the above mentioned Lyapunov function is the free energy F(ρ) := U(ρ) − TS(ρ) =

Z

Γ¡Hρ + Tρ logρ¢ dV . (2.43)

Remark 2.2. The internal energy U is defined as the mean value of the Hamiltonian with respect to the measure dµ = ρdV , i.e. U = 〈H〉_µ. On the other hand, the entropyS, whose actual form of is suggested by its canonical expression, is not a mean value, which is also stressed by its nonlinear dependence onρ.

Notice that the free energy F(ρ) is defined on the space of probability densities ρ such that R

ΓρdV = 1. For such a reason the differential of F at ρ in the arbitrary direction h must be computed respecting the latter normalization constraint, which can be done (for example) by the method of Lagrange multipliers. The latter method amounts to differentiate the function

F(e ρ,λ) := F(ρ) + λ µZ

Γρ dV − 1

¶

(2.44) with respect to both the densityρ and the parameter λ. The total differential ofF at (e ρ,λ) in the direction (h, dλ) is

dF(e ρ,λ; h,dλ) = dF(ρ; h) + λ Z

ΓhdV + dλ µZ

Γρ dV − 1

¶

=

= Z

Γ[H + T(logρ + 1) + λ]h dV + dλ µZ

Γρ dV − 1

¶ ,

and turns out to be zero for any direction (h, dλ) iff

½ R

ΓρdV = 1

H + T(logρ + 1) + λ = 0 ,

which implies ρ = ρeq = e^−H/T/Z. Thus, the only critical point of the free energy F is the Gibbs canonical density. One could also avoid to use the method of Lagrange multipliers, differentiating only F and taking into account the normalization constraint by observing that R

ΓhdV = 0 (ρ + ²h must be a density), which leads to R

Γ[H + T(logρ + 1)]hdV = 0; the latter condition is satisfied iff the square bracket is constant.

In order to study the nature of the critical densityρeq and decide whether it is a minimum or a maximum of F, one computes

F(ρeq+ ²h) − F(ρeq) = ² Z

Γ[H + T(logρeq+ 1)]h dV +²² 2T h²

ρeq dV +O⁽²³) =

= ²T(1 − log Z) Z

Γh dV +²² 2 T

Z

Γ

h²

ρeq dV +O⁽²³) =

= ²² 2T

Z

Γ

h² ρeq

dV +O⁽²³), (2.45)

where the linear term (with respect to ²h) disappears since R_ΓhdV = 0. One thus finds that, if² is small enough, the above variation is positive, so that ρeq is a local minimum of F. One also concludes that ρeq is actually the absolute minimum of F since the latter function does not admit other critical points.

One could be disappointed by the occurrence ofρeq in the denominator of (2.45), though this is actually a false problem. Indeed, one easily realizes that theO⁽²³) remainder is actually a O⁽²³h³/ρ²_eq). Thus, by the change of variable h = ρeqr, with the condition R

ΓrρeqdV = 0, one gets

F(ρeq(1 + ²r)) − F(ρeq) =²² 2T

Z

Γr²ρeq dV +O⁽²³r³ρeq),

which is well defined. In particular, the existence of the second differential d²F(ρeq; r) :=

d²F(ρeq(1 + ²r))/d²²|_²=0= TR

Γr²ρeqdV is ensured by r ∈L₂⁰⁽Γ^,µeq), the latter being the space of function square integrable with respect to the equilibrium measure dµeq:= ρeqdV , with zero mean.

Approach to equilibrium

In order to study the approach to equilibrium, i.e. the convergence ρ → ρeq, we study the time derivative of F(ρ(t)). We are going to show that ˙F ≤ 0, so that F is a non-increasing function of time. Moreover, it will be shown that the vector field of the Fokker-Planck equation is transversal to the manifold ˙F = 0, which implies that F decreases almost monotonically down to its absolute minimum value, which is necessarily reached at the unique critical point ρ = ρeq.

2.4. APPROACH TO EQUILIBRIUM I: NOISE & DISSIPATION 35 The time derivative of the free energy (2.43), computed on the solution of the Fokker-Planck equation (2.36), is given by

dF

where, in the second step (first line), the integral TR

Γ∂ρ/∂t vanishes since R_Γρ = 1, and in the last step the following quantities have been defined:

A := − On supposing that bothρ and ∇pρ vanish sufficiently fast as (q, p) approaches ∂Γ, repeatedly making use of the divergence theorem and of the relation T = D/γ, one finds the following.

A = −

It thus turns out that ˙F = B + C, namely dF

dt = −γ Z

Γ

¯

¯∇pH + T∇plogρ¯¯²ρ dV . (2.50) The latter relation implies that ˙F ≤ 0, i.e. F does not increase along the flow of the Fokker-Planck equation (2.36). In particular, F is stationary, i.e. ˙F = 0 holds, iff

∇pH + T∇plogρ = 0 ,

which is solved by lifting a ∇p, obtaining H + T logρ = −Φ(q, t), whereΦ(q, t) is an integration constant. One thus finds that the zeroes of ˙F(ρ) constitute an infinite-dimensional manifold M_F of canonical-like densities, namely

M_F:= {ρ : ˙F(ρ) = 0} =

½ ρ = 1

Z ^e−

H+ΦT , Φ^{(q, t) : Z}

ΓρdV = 1

¾

. (2.51)

We now show that the vector field (i.e. the right hand side) of the Fokker-Planck equation (2.36) cannot be parallel, and so is transversal to the manifold M_F, which implies that, if at time ¯t ˙F = 0 and ρ 6= ρeq, then at time ¯t + 0⁺ F < 0. The Fokker-Planck equation (2.36) reads˙

∂ρ/∂t = XF P(ρ), where

X_{F P}(ρ) := −{ρ, H} + ∇p· (γρ∇pH + D∇pρ) .

A one-parameter family of potentials λ 7→Φ^{(q, t;}λ), determines a curve on MF, namely λ 7→

ρ(λ) = e^−H+ΦT /Z⁽λ). As a consequence, the “vector”

∂ρ

∂λ = −ρ T

µ∂Φ

∂λ −

¿∂Φ

∂λ À¶

,

where 〈·〉 =R (·)ρdV , is tangent to MF atρ(λ). On the other hand, one easily checks that X_{F P}(ρ(λ)) = ρ

T{Φ, H} = ρ

T∇qΦ· ∇pH .

The latter is tangent to M_F if it is parallel to the tangent vector∂ρ/∂λ, namely if

∂Φ

∂λ −

¿∂Φ

∂λ À

= c¡∇qΦ· ∇pH¢

for some constant c 6≡ 0. The latter relation cannot hold, in general, since the left hand side de-pends only on (q, t), whereas the right hand side dede-pends also on p (unless H dede-pends linearly on p, which is generically excluded). Thus, the only way of satisfying the above parallelism condition is that both sides identically vanish, independently of each other, which implies that Φdepends only on t, and, as a consequence, ρ = ρeq (show that).

The conclusion of the above argument is that F(ρ) → min F = F(ρeq) as t → +∞. However, one cannot simply conclude that ρ → ρeq, which is meaningful only when the norm ruling the distances is specified. Notice that, due to some technical reasons, in infinite-dimensional problems the Lyapunov function method does not ensure stability, in general. In the present case, one can prove, for example, that if the temperature T = D/γ is large enough, and in the limit N → ∞,R

Γ(ρ − ρeq)²dV → 0 exponentially fast.

Nel documento HAMILTONIAN MECHANICS —————- (pagine 32-37)