Unified formalism for entropy production and fluctuation relations

Abstract

Stochastic entropy production, which quantifies the difference between the probabilities of trajectories of a stochastic dynamics and its time reversals, has a central role in nonequilibrium thermodynamics. In the theory of probability, the change in the statistical properties of observables due to reversals can be represented by a change in the probability measure. We consider operators on the space of probability measure that induce changes in the statistical properties of a process, and formulate entropy productions in terms of these change-of-probability-measure (CPM) operators. This mathematical underpinning of the origin of entropy productions allows us to achieve an organization of various forms of fluctuation relations: All entropy productions have a non-negative mean value, admit the integral fluctuation theorem, and satisfy a rather general fluctuation relation. Other results such as the transient fluctuation theorem and detailed fluctuation theorems then are derived from the general fluctuation relation with more constraints on the operator of a entropy production. We use a discrete-time, discrete-state-space Markov process to draw the contradistinction among three reversals of a process: time reversal, protocol reversal and the dual process. The properties of their corresponding CPM operators are examined, and the domains of validity of various fluctuation relations for entropy productions in physics and chemistry are revealed. We also show that our CPM operator formalism can help us rather easily extend other fluctuations relations for excess work and heat, discuss the martingale properties of entropy productions, and derive the stochastic integral formulas for entropy productions in constant-noise diffusion process with Girsanov theorem. Our formalism provides a general and concise way to study the properties of entropy-related quantities in stochastic thermodynamics and information theory. *

Improved bounds on entropy production in living systems

“…As our theoretical analysis assumed a NESS, the thermodynamic interpretation of the bounds on σ requires us to revisit precisely whether or not this assumption is valid under relevant experimental conditions. While entropy can be defined more generally along a single stochastic trajectory (17,89), the average rate of entropy production in a NESS represents the rate of entropy increase in the chemical or thermal reservoirs coupled to the system, which drive the NESS. For isothermal systems, σ = Q hk /T , where Q hk is the housekeeping heat (35,90), which quantifies the rate at which heat is dissipated into the environment.…”

Stochastic thermodynamics of relativistic Brownian motion

“…From a thermodynamic point of view, these cosmic jets are, thus, nothing else but ensembles of Brownian particles that are driven very far from thermal equilibrium. Therefore, it seems plausible that recent advances in stochastic thermodynamics [2][3][4][5][6][7] should be useful in describing the thermal properties also of cosmic jets. However, with the exception of the fluctuation theorem derived by Fingerle [8], none of the existing results appear directly applicable at relativistic energies.…”

Thermodynamics of Gambling Demons

“…We derive these results applying the theory of Martingale stochastic processes. Martingales have been fruitfully applied in probability theory [12], quantitative finance [13], and more recently in nonequilibrium thermodynamics [14][15][16][17][18], providing insights beyond standard fluctuation theorems, e.g., universal bounds for the extrema and stopping-time statistics of thermodynamic quantities [14,15,[19][20][21][22].…”