Entropy Anomaly in Langevin-Kramers Dynamics with Matrix Drag and Diffusion
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
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
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
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
where and .
It is often convenient to define and write the SDE in the equivalent form
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”.
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.
For any , we have
as , where is the solution the SDE
, called the noise induced drift, is an anomalous drift term that arises in the limit. It is given by
for all , .
Note that we define the components of such that
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
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
Let , , and be a function that satisfies
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:
When the vector potential, , vanishes we have the simplified result, Corollary 2. As ,
The paper concludes with Section 5.1, where we use a heuristic argument to isolate the anomalous entropy contribution,
i.e. the difference in the entropy production between the under and overdamped equations in the small mass limit.
2 Background: Time Reversal and Entropy Production
2.1 Time Inversion
Consider a generic SDE in Stratonovich form
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
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
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
where , is the pseudoinverse, denotes the Stratonovich integral, and
3 Time Inversion in Langevin Dynamics
In this section, we consider time inversion for dissipative Langevin dynamics,
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
Note that contains the dissipative component of the dynamics, as discussed above.
with Hamiltonian Eq. (2).
3.2 Time Inversion for the Overdamped Limit
Theorem 1.1 gives the small mass limit of the forward and backward processes respectively:
where and are computed via Eq. (17) using the vector potentials and respectively.
The natural spacetime inversion for the overdamped dynamics is
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
Using Eq. (28), along with the assumption that the noise only couples to the momentum, we find
The fluctuation dissipation relation, Eq. (10), yields
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
Next, we use the form of the Hamiltonian Eq. (2), along with an additional assumption.
For the remainder of this work, we assume is independent of .
Recalling , the entropy produced in the environment can be written as
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
As in BirrellHomogSDE (), Itô’s formula can be used to compute
where we define
Our strategy for homogenizing processes of the form is to find a function and a function such that
By formally applying the Fredholm alternative, one is led to the ansatz
where solves with .
We will be able to make this motivating discussion rigorous under the following additional assumptions.
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 ,