Classical and quantum stochastic thermodynamics

Abstract

The stochastic thermodynamics provides a framework for the description of systems that are out of thermodynamic equilibrium. It is based on the assumption that the elementary constituents are acted by random forces that generate a stochastic dynamics, which is here represented by a Fokker-Planck-Kramers equation. We emphasize the role of the irreversible probability current, the vanishing of which characterizes the thermodynamic equilibrium and yields a special relation between fluctuation and dissipation. The connection to thermodynamics is obtained by the definition of the energy function and the entropy as well as the rate at which entropy is generated. The extension to quantum systems is provided by a quantum evolution equation which is a canonical quantization of the Fokker-Planck-Kramers equation. An example of an irreversible systems is presented which shows a nonequilibrium stationary state with an unceasing production of entropy. A relationship between the fluxes and the path integral is also presented.

Kinetic theory of gases: stochastic thermodynamics avant la lettre

“…After that, we show that the probabilistic and stochastic reasoning they used can be employed to derive a Kolmogorov equation associated with a stochastic dynamics [21,22]. These results allow us to say that the kinetic theory can be understood as stochastic thermodynamics [28][29][30][31][32] avant la lettre.…”

“…In this approximation, the master equation is recognized as the Boltzmann equation. It should be remarked that the original Boltzmann equation, given by (32), is written in terms of velocities only. This happens because the transition probability rate considered by Boltzmann, as well as by Maxwell, depends only on velocities of the molecules and not on their positions.…”

Optimization and Efficiency Studies of Heat Engines: A Review

“…This general multi-parameter combined objective function can be adopted as the optimization criterion instead of the single optimization criterion to derive the universal expression of the efficiency for a simple model of heat engines In Ref. [151], by using stochastic thermodynamic analysis with a master equation description of a driven open system 155 , the universal expression of the optimization efficiency with a multi-parameter combined objective function is derived.…”

Explorando O Calor Na Termodinâmica Estocástica

“…O primeiro método, é o algorítimo de Euler-Maryuama [77]. Considere a equação de Langevin genérica ẋ = F (x) + σζ(t), (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16) onde σ é uma constante, e ζ(t) é um ruído branco Gaussiano branco, que satisfaz ⟨ζ(t)⟩ = 0 e ⟨ζ(t)ζ(t ′ )⟩ = δ(t − t ′ ). O algorítimo consiste em fornecer um valor inicial x(0), fornecer a variável aleatória ζ(t) e calcular o incremento ∆x(0) = F (x(0)) + σζ(0), (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17) de tal forma que, o próximo instante de tempo, h, vai ser dado por…”

“…Inicialmente, os gregos pensavam que o calor estava associado ao fogo, mas foi somente na era moderna que o calor foi associado ao movimento das moléculas [14]. Atualmente, entendemos o calor como apenas uma forma de transferência de energia entre muitas outras [6,15]. Sua característica distintiva é ser uma transferência de energia desordenada, que está quase sempre presente em qualquer processo termodinâmico.…”