Entropy Anomaly in Langevin-Kramers Dynamics with Matrix Drag and Diffusion

# Entropy Anomaly in Langevin-Kramers Dynamics with Matrix Drag and Diffusion

## Abstract

We investigate entropy production in the small mass (or overdamped) limit of Langevin-Kramers dynamics. Our results apply to systems with magnetic field as well as matrix valued drag and diffusion coefficients that satisfy a version of the fluctuation dissipation relation with state dependent temperature. In particular, we generalize the anomalous entropy production results of PhysRevLett.109.260603 ().

As a part of this work, we develop a theory for homogenizing a class of integral processes involving the position and scaled velocity variables. This allows us to rigorously prove convergence of the entropy produced in the environment, including a bound on the convergence rate.

###### Keywords:
stochastic Langevin equation small mass limit homogenization anomalous entropy production
60H10 82C31

## 1 Introduction

Langevin-Kramers equations model the motion of a noisy, damped, diffusing particle of non-zero mass, . In the simplest case, the stochastic differential equation (SDE) has the form

 dqt=vtdt,mdvt=−γvtdt+σdWt, (1)

where and are the dissipation (or drag) and diffusion coefficients respectively and is a Wiener process. Smoluchowski smoluchowski1916drei () and Kramers KRAMERS1940284 () pioneered the study of such diffusive systems in the small mass (or overdamped) limit; see Nelson1967 () for more on the early literature and doi:10.1137/S1540345903421076 (); Chevalier2008 (); bailleul2010stochastic (); pinsky1976isotropic (); pinsky1981homogenization (); Jorgensen1978 (); dowell1980differentiable (); XueMei2014 (); angst2015kinetic (); bismut2005hypoelliptic (); bismut2015 () for further studies.

Chetrite and Gawȩdzki Chetrite2008 (); gawedzki2013fluctuation () have developed a theory of time reversal and entropy production in SDEs, which we summarize in Section 2. In PhysRevLett.109.260603 () it was shown that the overdamped (i.e. small mass) limit of Langevin-Kramers dynamics exhibits an entropy anomaly; the entropy production associated with the limiting overdamped SDE has a deficit when compared to the small mass limit of the entropy produced by the underdamped SDE. This effect was shown to arise in systems with a nonzero temperature gradient.

In this paper, we generalize the study of Langevin-Kramers entropy production and the entropy anomaly to systems with magnetic field and state dependent matrix-valued drag and diffusion. Specifically, we prove a rigorous convergence result and convergence rate bound for the entropy produced in the environment.

### 1.1 Previous Results

The Hamiltonian of a particle of mass and charge in an electromagnetic field with -vector potential and -electrostatic potential is

 H(t,x)=12m∥p−eϕ(t,q)∥2+eU(t,q) (2)

where . Allowing for an additional continuous forcing term, , and coupling to noise and linear drag via the -matrix valued functions and respectively, Hamilton’s equations for this system are

 dqmt= 1m(pmt−ψ(t,qmt))dt, (3) d(pmt)i= (−1mγij(t,qmt)δjk((pmt)k−ψk(t,qmt))+~Fi(t,xmt)−∂qiV(t,qmt) (4) +1m∂qiψk(t,qmt)δjk((pmt)j−ψj(t,qmt)))dt+σiρ(t,qmt)dWρt,

where and .

It is often convenient to define and write the SDE in the equivalent form

 dqmt= 1mumtdt, (5) d(umt)i= (−1m~γik(t,qmt)(umt)kdt+Fi(t,xmt))dt+σiρ(t,qmt)dWρt, (6)

where

 ~γik(t,q)≡γik(t,q)+Hik(t,q)≡γik(t,q)+∂qkψi(t,q)−∂qiψk(t,q), (7)

and

 F(t,x)=−∂tψ(t,q)−∇qV(t,q)+~F(t,x). (8)

Here and in the following we employ the summation convention for repeated indices.

In this paper we will assume the fluctuation dissipation relation holds pointwise for a time and state dependent “temperature”.

###### Assumption 1

Define

 Σij(t,q)=∑ρσiρ(t,q)σjρ(t,q). (9)

We assume

 Σ(t,q)=2β−1(t,q)γ(t,q), (10)

where is a function that is bounded above and below by positive constants. Physically, is related to the time and position dependent “temperature” by , where is Boltzmann’s constant.

In BirrellHomogenization () it was shown that, for a large class of such systems, there exists unique global in time solutions that converge to as , where here is the solution to a certain limiting SDE. We summarize the precise mode of convergence in Theorem 1.1 below, which we take as the starting point for this work. See Appendix A for a list of properties that guarantee that the following result holds.

###### Theorem 1.1

For any , we have

 supt∈[0,T]E[∥umt∥p]1/p=O(m1/2),supt∈[0,T]E[∥qmt−qt∥p]1/p=O(m1/2) (11)

as , where is the solution the SDE

 dqt= ~γ−1(t,qt)F(t,qt,ψ(t,qt))dt+S(t,qt)dt+~γ−1(t,qt)σ(t,qt)dWt. (12)

, called the noise induced drift, is an anomalous drift term that arises in the limit. It is given by

 Si(t,q)≡β−1(t,q)∂qj(~γ−1)ij(t,q). (13)

also satisfies

 E[supt∈[0,T]∥qt∥p]1/p<∞ (14)

for all , .

Note that we define the components of such that

 (~γ−1)ij~γjk=δik, (15)

and for any we define the contraction .

The study of the singular nature of the Langevin-Kramers system in the small mass limit (i.e. the appearance of the noise induced drift) has a long history PhysRevA.25.1130 (); Sancho1982 (); volpe2010influence (); Hottovy2014 (); herzog2015small (); particle_manifold_paper (); BirrellHomogenization (). See Hottovy2014 () for further references and discussion.

The assumptions that and are and is allow us to rewrite the limiting equation in Stratonovich form

 dqt= ~γ−1(t,qt)F(t,qt,ψ(t,qt))dt+~S(t,qt)dt+~γ−1(t,qt)σ(t,qt)∘dWt, (16)

where

 ~Si(t,q)= β−1(t,q)∂qj(~γ−1)il(t,q)(~γ−1)jk(t,q)Hkl(t,q) (17) −12∑ξ(~γ−1)il(t,q)∂qkσlξ(t,q)(~γ−1(t,q)σ(t,q))kξ.

This form of the equation will be useful for our subsequent discussion of entropy production.

In addition, 2017arXiv170505004B () contains a convergence result for the joint distribution of , where

 zmt≡umt/√m. (18)

The properties in Appendix A, along with Assumption 1, imply the following.

###### Theorem 1.2

Let , , and be a function that satisfies

 ∥∇~h(q,z)∥≤K(1+∥(q,z)∥q). (19)

Define

 Ht=E⎡⎢⎣(β(t,qt)2π)n/2∫~h(qt,z)e−β(t,qt)∥z∥2/2dz⎤⎥⎦. (20)

Then

 E[~h(qmt,zmt)]=Ht+O(mδ) (21)

as .

This paper will build on these prior convergence results to study the entropy production in the small mass limit.

### 1.2 Summary of Results

Our main result, Theorem 4.3, is a formula for the entropy produced in the environment for the Langevin-Kramers system with non-zero magnetic field and matrix valued drag and diffusion. The general result is found in Theorem 4.3 and holds under the following conditions:

• The fluctation dissipation relation, Eq. (10), holds.

• The properties from Appendix A hold.

• is independant of

• and are .

• For any the following are polynomially bounded in , uniformly in : , , , , , , , , , , , , , , and . i.e. there exists , such that

 supt∈[0,T]|∂tβ(t,q)|≤~C(1+∥q∥~p) (22)

and so on.

When the vector potential, , vanishes we have the simplified result, Corollary 2. As ,

 E[Senv,ms,t] (23) = +∫tsE[β−1(r,qr)∇q⋅(γ−1(V∇qβ+β~F))(r,qr)]dr +n+22E[ln(β(t,qt)/β(s,qs))]−E[∫ts(β−1∂rβ)(r,qr)dr] +∫tsE[((−∇qV+~F)⋅γ−1(V∇qβ+β~F))(r,qr)]dr +∫tsE[(β−3∇qβ⋅(3n+26γ−1−∫∞0Tr[γe−2yγ]γ−1e−yγdy)∇qβ)(r,qr)]dr +O(mδ).

The paper concludes with Section 5.1, where we use a heuristic argument to isolate the anomalous entropy contribution,

 E[Sanoms,t] (24) = ∫tsE[(β−3∇qβ⋅(3n+26γ−1−∫∞0Tr[γe−2yγ]γ−1e−yγdy)∇qβ)(r,qr)]dr.

i.e. the difference in the entropy production between the under and overdamped equations in the small mass limit.

As our main technical tools, we prove two homogenization results about processes of the form in the limit . See Theorems 4.1 and 4.2.

## 2 Background: Time Reversal and Entropy Production

In this section we present a synopsis of the theory of time reversal and entropy production, as developed by Chetrite and Gawȩdzki Chetrite2008 (); gawedzki2013fluctuation ().

### 2.1 Time Inversion

Consider a generic SDE in Stratonovich form

 dxt=b(t,xt)dt+~σ(t,xt)∘dWt (25)

on the time interval , driven by a Wiener process and smooth drift and diffusion .

A time inversion on spacetime will be given by a map where (which we will also write as ) is a smooth involution and . We will be primarily interested in the case where with position and momentum components and respectively, and

 (q,p)∗=(q,−p), (26)

but we keep the discussion general for now.

One could define the time reversed trajectories of the original system Eq. (25) by , however this is problematic as, for example, it leads to anti-dissipation. A more physically reasonable method of defining the time reversed dynamics is to split the drift into two components (called the dissipative and conservative parts, respectively Chetrite2008 ()) and define the time reversed process to be the solution to the SDE

 dx′t=(ϕ∗b+)(t∗,x′t)dt−(ϕ∗b−)(t∗,x′t)dt±(ϕ∗~σ)(t∗,x′t)∘dWt, (27)

where denotes the pushforward of vector fields by the smooth map . The choice of on the noise term doesn’t impact the distribution of the process and so can be chosen based on convenience. We call the solution the backward process while will be called the forward process.

For our purposes, the splitting of , and the corresponding sign change for the component in the backward equation, will be chosen so that the dissipative component of the original SDE remains dissipative in the SDE for the backward process.

### 2.2 Entropy Production

The entropy produced by the process Eq. (25) is defined via the Radon-Nikodym derivative of the distribution of the backward process w.r.t. the forward process; ee Chetrite2008 () for details. In particular, the entropy produced in the environment from time to time , can be computed via the formula

 Senvs,t≡ ∫ts2^bj+(r,xr)(~Σ−1)jk(r,xr)∘dxkr (28) −∫ts2^bj+(r,xr)(~Σ−1)jk(r,xr)bk−(r,xr)+∇⋅b−(r,xr)dr,

where , is the pseudoinverse, denotes the Stratonovich integral, and

 ^bi+=bi+−12∑ξ~σiξ∂j~σjξ. (29)

## 3 Time Inversion in Langevin Dynamics

In this section, we consider time inversion for dissipative Langevin dynamics,

 dxt=(−Γ(t,xt)∇Ht(xt)+Π(t,xt)∇Ht(xt)+Gt(xt))dt+~σ(t,xt)∘dWt, (30)

where is a time dependent Hamiltonian function, is an antisymmetric matrix, is a symmetric, positive semidefinite matrix, is an additional non-conservative force field, and are the noise coefficients. We emphasize that it will be important for us that the equation is given is Stratonovich form.

After discussing time inversion, we derive a simplified formula for the entropy produced in the environment, Eq. (28) by an inertial particle in both the under and overdamped regimes.

### 3.1 Time Inversion for Langevin-Kramers Dynamics

Chetrite and Gawedzki Chetrite2008 () define the canonical splitting of the drift in Eq. (30) by

 b+=−Γt∇Ht,b−=Π∇Ht+Gt. (31)

Note that contains the dissipative component of the dynamics, as discussed above.

We specialize to Langevin-Kramers dynamics (i.e. an underdamped inertial particle) with the time inversion map Eq. (26). Specifically, we let and assume the objects in Eq. (30) have the form

 Γ(t,x)= (000γ(t,q)),~σ(t,x)=(000σ(t,q)), (32) Π(t,x)= (0I−I0),G(t,x)=(0,~F(t,x)), (33)

with Hamiltonian Eq. (2).

In this case, we have

 b+(t,x)= (0,−1mγ(t,q)(p−ψ(t,q))), (34) b−(t,x)= (1m(p−ψ(t,q)),1mδij(pi−ψi(t,q))∇qψj(t,q)−∇qV(t,q)+~F(t,x)). (35)

The pushforwards are

 (ϕ∗b+)(t,x)=(0,1mγ(t,q)(−p−ψ(t,q))), (36) (ϕ∗b−)(t,x) (37) = (1m(−p−ψ(t,q)),−1mδij(−pi−ψi(t,q))∇qψj(t,q)+∇qV(t,q)−~F(t,q,−p)).

Hence the time reversed dynamics are given by

 dq′t= 1m(p′t+ψ(t∗,q′t))dt, (38) d(p′t)i= −1mδjk(γij(t∗,q′t)+∂qiψj(t∗,q′t))(p′t+ψ(t∗,q′t))kdt (39) +(−∂qiV(t∗,q′t)+~Fi(t∗,q′t,−p′t))dt+σ(t∗,q′t)∘dWt.

Note that these equations have the same form as the original system, Eq. (3)-Eq. (4), but the explicit time dependence is reversed and the vector potential has its sign reversed.

### 3.2 Time Inversion for the Overdamped Limit

Theorem 1.1 gives the small mass limit of the forward and backward processes respectively:

 dqt= ~γ−1(t,qt)(−∂tψ(t,qt)−∇qV(t,qt)+~F(t,qt,ψ(t,qt)))dt (40) +~S(t,qt)dt+~γ−1(t,qt)σ(t,qt)∘dWt,
 dq′t= (~γT)−1(t∗,q′t)(−∂tψ(t∗,q′t)−∇qV(t∗,q′t)+~F(t∗,q′t,ψ(t∗,q′t)))dt (41) +~S′(t∗,q′t)dt+(~γT)−1(t∗,q′t)σ(t∗,q′t)∘dWt.

where and are computed via Eq. (17) using the vector potentials and respectively.

The natural spacetime inversion for the overdamped dynamics is

 (t,q)→(t∗,q). (42)

In the case of nontrivial , the limiting forwards and backwards equations do not correspond to one another under any time inversion rule of the form Eq. (27), as the noise terms differ by more than just a replacement . However, if then they do correspond under the rule , , called in Chetrite2008 () the reversed protocol.

We therefore proceed in two steps. First, in Section 4 we investigate the entropy production in the environment for the underdamped system and derive a formula for its small mass limit in the case of non-zero .

We then specialize to , in which case we can directly compute the entropy production in the environment for the overdamped system and compare it to the limit of the underdamped system. A formal calculation will then result in a formula for the total entropy production in each case, and we will find that the results differ i.e. the operations of computing the entropy production and taking the small mass limit do not commute. This anomalous entropy production was first derived formally in PhysRevLett.109.260603 (). Our treatment puts one aspect of this derivation on a rigorous footing, the convergence of the entropy produced in the environment, including an explicit convergence rate bound. It also generalizes the derivation by allowing for matrix-valued and that are related via the fluctuation dissipation relation, Eq. (10).

## 4 Entropy Production for Underdamped Langevin-Kramers Dynamics

In this section we derive a formula for the entropy production that results from the inversion rule Eq. (26) for the underdamped system from Section 3.1.

Using Eq. (28), along with the assumption that the noise only couples to the momentum, we find

 Senvs,t= −∫ts2∂plH(r,xr)γjl(r,qr)(Σ−1)jk(r,qr)∘d(pr)k +∫ts2∂plH(r,xr)γjl(r,qr)(Σ−1)jk(r,qr)(−∇qH+~F)k(r,xr)−∇p⋅~F(r,xr)dr.

The fluctuation dissipation relation, Eq. (10), yields

 Senvs,t= −∫tsβ(r,qr)∂pkH(r,xr)∘d(pr)k +∫tsβ(r,xr)∂pkH(r,xr)(−∇qH+~F)k(r,xr)−∇p⋅~F(r,xr)dr.

The fact that , , where the first index refers to the -variables and the second to the -variables, allows us to use Itô’s formula for the Stratonovich integral to obtain

 β(t,qt)H(t,xt)−β(s,qs)H(s,xs) (43) = ∫ts∂r(βH)(r,xr)dr+∫ts∇q(βH)(r,xr)⋅dqr+∫ts∇p(βH)(r,xr)∘dpr.

Therefore

 Senvs,t= −(β(t,qt)H(t,xt)−β(s,qs)H(s,xs))+∫ts∂r(βH)rdr (44) +∫tsHr∇qβr⋅∇pHrdr+∫tsβr∇qHr⋅∇pHrdr +∫tsβr∇pHr⋅(−∇qHr+~Fr)−∇p⋅~Frdr = −(β(t,qt)H(t,xt)−β(s,qs)H(s,xs))+∫ts∂r(βH)rdr (45) +∫ts(Hr∇qβr+βr~Fr)⋅∇pHr−∇p⋅~Frdr.

Next, we use the form of the Hamiltonian Eq. (2), along with an additional assumption.

###### Assumption 2

For the remainder of this work, we assume is independent of .

Recalling , the entropy produced in the environment can be written as

 Senv,ms,t= −(β(t,qmt)H(t,qmt,zmt)−β(s,qms)H(s,qms,zms))+∫ts∂r(βV)(r,qmr)dr +12∫ts∂rβ(r,qmr)∥zmr∥2dr+12√m∫ts∥zmr∥2∇qβ(r,qmr)⋅zmrdr +1√m∫ts((V∇qβ)(r,qmr)+(β~F)(r,qmr)−(β∂rψ)(r,qmr))⋅zmrdr, (46)

where .

### 4.1 Homogenization of Integral Processes

In this section, we develop the techniques necessary to investigate the entropy production in the underdamped system, Eq. (4), in the limit .

The terms in Eq. (4) of the form converge in distribution by Theorem 1.2. The term will be shown to converge by using Theorem Eq. (1.1) (i.e. because ). That leaves investigating the convergence of the integral terms involving as our primary task.

General homogenization results about the limit of integral processes of the form , where come from solving some family of Hamiltonian system parametrized by (analogous to ), can be found in BirrellHomogSDE (). The situation here is substantially simpler than the general case, so we reproduce a streamlined version of that argument here.

As a starting point, let be , meaning is and, for each , is in with second derivatives continuous jointly in all variables.

Define the operator and its formal adjoint, , by

 (Lχ)(t,q,z)= 12Σkl(t,q)(∂zk∂zlχ)(t,q,z)−~γkl(t,q)δlizi(∂zkχ)(t,q,z), (47) (L∗h)(t,q,z)= ∂zk(12Σkl(t,q)∂zlh(t,q,z)+~γkl(t,x)δlizih(t,q,z)). (48)

As in BirrellHomogSDE (), Itô’s formula can be used to compute

 ∫ts(Lχ)(s,qmr,zmr)dr (49) = m1/2(Rm1)s,t+m(χ(t,qmt,zmt)−χ(s,qms,zms)+(Rm2)s,t),

where we define

 (Rm1)s,t= −∫ts(∇qχ)(r,qmr,zmr)⋅zmrdr (50) −∫ts(∇zχ)(r,qmr,zmr)⋅[(~F(r,qmr)−∇qV(r,qmr))dr+σ(r,qmr)dWr],

and

 (Rm2)s,t= −∫ts∂rχ(r,qmr,zmr)dr. (51)

Our strategy for homogenizing processes of the form is to find a function and a function such that

 Lχ=G−~G (52)

(a variant of the cell problem from pavliotis2008multiscale () ). Assuming is chosen so that doesn’t grow too fast in , we will be able to use Eq. (49) to prove

 ∫tsG(r,qmr,zmr)dr→∫ts~G(r,qr)dr. (53)

By formally applying the Fredholm alternative, one is led to the ansatz

 ~G(t,q)=∫h(t,q,z)G(t,q,z)dz. (54)

where solves with .

We will be able to make this motivating discussion rigorous under the following additional assumptions.

###### Assumption 3

From this point on, we assume:

• The properties from Appendix A hold.

• and are .

• For any the following are polynomially bounded in , uniformly in : , , , , , , , , , , , , , , , and . i.e. there exists ,