Stochastic Energetics of Non-uniform Temperature Systems

Journal of Statistical Physics, 1982

A suitable extension of the Mori memory-function formalism to the non-Hermitian case allows a "multiplicative" process to be described by a Langevin equation of non-Markoffian nature. This generalized Langevin equation is then shown to provide for the variable of interest the same autocorrelation function as the well-known theoretical approach developed by Kubo, the stochastic Liouville equation (SLE) theory. It is shown, furthermore, that the present approach does not disregard the influence of the variable of interest on the time evolution of its thermal bath. The stochastic process under study can also be described by a Fokker-Planck-like equation, which results in a Gaussian equilibrium distribution for the variable of interest. The main flaw of the SLE theory, that resulting in an uncorrect equilibrium distribution, is therefore completely eliminated.

Physical Review E, 2009

Non-Markovian stochastic Langevin-like equations of motion are compared to their corresponding Markovian (local) approximations. The validity of the local approximation for these equations, when contrasted with the fully nonlocal ones, is analyzed in details. The conditions for when the equation in a local form can be considered a good approximation are then explicitly specified. We study both the cases of additive and multiplicative noises, including system dependent dissipation terms, according to the Fluctuation-Dissipation theorem.

Journal of Mathematical Physics, 1977

It is shown that all statistical properties of the generalized Langevin equation with Gaussian fluctuations are determined by a single, two-point correlation function. The resulting description corresponds with a stationary, Gaussian, non-Markovian process. Fokker-Planck-like equations are discussed, and it is explained how they can lead one to the erroneous conclusion that the process is nonstationary, Gaussian, and Markovian.

PHYSICAL REVIEW RESEARCH 4, 033247 (2022), 2022

Using the recently developed covariant Ito-Langevin dynamics, we develop a nonequilibrium thermodynamic theory for small systems coupled to multiplicative noises. The theory is based on Ito calculus, and is fully covariant under time-independent nonlinear transformation of variables. Assuming instantaneous detailed balance, we derive expressions for various thermodynamic functions, including work, heat, entropy production, and free energy, both at ensemble level and at trajectory level, and prove the second law of thermodynamics for arbitrary nonequilibrium processes. We relate time-reversal asymmetry of path probability to entropy production, and derive its consequences such as fluctuation theorem and nonequilibrium work relation. For Langevin systems with additive noises, our theory is equivalent to the common theories of stochastic energetics and stochastic thermodynamics. Using concrete examples, we demonstrate that whenever kinetic coefficients or metric tensor depend on system variables, the common theories of stochastic thermodynamics and stochastic energetics should be replaced by our theory.

arXiv: Mathematical Physics, 2017

We study a class of systems whose dynamics are described by non-Markovian Langevin equations with state-dependent coefficients. We find that in the limit, in which all the characteristic time scales vanish at the same rate, the position variable of the system converges to a homogenized process, described by an equation containing additional drift terms induced by the noise in the considered limit. The convergence results are obtained using the main result in \cite{Hottovy2014}, proven here in a stronger version, requiring weaker spectral assumption on the damping matrix. We apply our results to study thermophoresis of a Brownian particle in a non-equilibrium heat bath.

Journal of Statistical Mechanics: Theory and Experiment, 2009

The non-linear dissipation corresponding to a non-Gaussian thermal bath is introduced together with a multiplicative white noise source in the phenomenological Langevin description for the velocity of a particle moving in some potential landscape. Deriving the closed Kolmogorov's equation for the joint probability distribution of the particle displacement and its velocity by use of functional methods and taking into account the well-known Gibbs form of the thermal equilibrium distribution and the condition of 'detailed balance' symmetry, we obtain the exact master equation: given the white noise statistics, this master equation relates the non-linear friction function to the velocitydependent noise function. In particular, for multiplicative Gaussian white noise this operator equation yields a unique inter-relation between the generally nonlinear friction and the (multiplicative) velocity-dependent noise amplitude. This relation allows us to find, for example, the form of velocity-dependent noise function for the case of non-linear Coulomb friction.

Physica B-Condensed Matter, 2004

The stochastic Langevin Landau-Lifshitz equation is usually utilized in micromagnetics formalism to account for thermal effects. Commonly, two different interpretations of the stochastic integrals can be made: Ito and Stratonovich. In this work, the Langevin-Landau-Lifshitz (LLL) equation is written in both Cartesian and Spherical coordinates. If Spherical coordinates are employed, the noise is additive, and therefore, Ito and Stratonovich solutions are equal. This is not the case when (LLL) equation is written in Cartesian coordinates. In this case, the Langevin equation must be interpreted in the Stratonovich sense in order to reproduce correct statistical results.

Probability Theory and Related Fields, 1999

We consider the stochastic heat equation in one space dimension and compute-for a particular choice of the initial datum-the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds to the identification of the sample Lyapunov exponent. Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion process.

2009

The average thermodynamic power of a time-dependent external potential in the white-noise Langevin model is derived using a Green's function solution. The power appears as a driving term in the differential equation for the average energy and determines whether the solution is stationary or non-stationary. Different dynamics are illustrated with explicit models: a linear potential with a static magnetic field, a linear potential perturbed with an oscillating component and a magnetic field switch modeled using a $\tanh$ protocol.

Physical review. E, 2020

In the present study we have developed an alternative formulation for the quantum stochastic thermodynamics based on the c-number Langevin equation for the system-reservoir model. This is analogous to the classical one. Here we have considered both Markovian and non-Markovian dynamics (NMD). Consideration of the NMD is an important issue at the current state of the stochastic thermodynamics. Applying the present formalism, we have carried out a comparative study on the heat absorbed and the change of entropy with time for a linear quantum system and its classical analog for both Markovian and NMD. Here the strength of the thermal noise and its correlation time for the respective cases are the leading quantities to explain any distinguishable feature which may appear with the equilibration kinetics. For another application, we have proposed a formulation with classical look for a quantum stochastic heat engine. Using it we have presented a comparative study on the efficiency and its ...