For a generic overdamped Langevin dynamics driven out of equilibrium by both time-dependent and nonconservative forces, the entropy production rate can be decomposed into two positive terms, termed excess and housekeeping entropy. However, this decomposition is not unique: There are two distinct decompositions, one due to Hatano and Sasa, the other one due to Maes and Netočny. Here, we establish the connection between these two decompositions and provide a simple, geometric interpretation. We show that this leads to a decomposition of the entropy production rate into three positive terms, which we call excess, housekeeping and coupling part, respectively. The coupling part characterizes the interplay between the time-dependent and nonconservative forces. We also derive thermodynamic uncertainty relations for the excess and housekeeping entropy in both the Hatano-Sasa and Maes-Netočny-decomposition and show that all quantities obey integral fluctuation theorems. We illustrate the decomposition into three terms using a solvable example of a dragged particle in a nonconservative force field.
We present a simple derivation of the integral fluctuation theorems for excess housekeeping heat for an underdamped Langevin system, without using the concept of dual dynamics. In conformity with the earlier results, we find that the fluctuation theorem for housekeeping heat holds when the steady state distributions are symmetric in velocity, whereas there is no such requirement for the excess heat. We first prove the integral fluctuation theorem for the excess heat, and then show that it naturally leads to the integral fluctuation theorem for housekeeping heat. We also derive the modified det…
We propose a housekeeping/excess decomposition of entropy production for general nonlinear dynamics in a discrete space, including chemical reaction networks and discrete stochastic systems. We exploit the geometric structure of thermodynamic forces to define the decomposition; this does not rely on the notion of a steady state, and even applies to systems that exhibit multistability, limit cycles, and chaos. In the decomposition, distinct aspects of the dynamics contribute separately to entropy production: the housekeeping part stems from a cyclic mode that arises from external driving, gener…
A nonequilibrium system is characterized by a set of thermodynamic forces and fluxes which give rise to entropy production (EP). We show that these forces and fluxes have an information-geometric structure, which allows us to decompose EP into contributions from different types of forces in general (linear and nonlinear) discrete systems. We focus on the excess and housekeeping decomposition, which separates contributions from conservative and nonconservative forces. Unlike the Hatano-Sasa decomposition, our housekeeping/excess terms are always well-defined, including in systems with odd varia…
Two qualitatively different ways of driving a physical system out of equilibrium, time-dependent and non-conservative forcing, are reflected by the decomposition of the system's entropy production into excess and housekeeping parts. We show that the difference between these two types of driving gives rise to a geometric formulation in terms of two orthogonal contributions to the currents in the system. This geometric picture in a natural way leads to variational expressions for both the excess and housekeeping entropy, which allow calculating both contributions independently from the trajector…
We investigate the decomposition of the total entropy production in continuous stochastic dynamics when there are odd-parity variables that change their signs under time reversal. The first component of the entropy production, which satisfies the fluctuation theorem, is associated with the usual excess heat that appears during transitions between stationary states. The remaining housekeeping part of the entropy production can be further split into two parts. We show that this decomposition can be achieved in infinitely many ways characterized by a single parameter σ. For an arbitrary value of …
Generalized free energy and excess/housekeeping decomposition in nonequilibrium systems: from large deviations to thermodynamic speed limits
ICREA-Complex Systems Lab, Universitat Pompeu Fabra（ICREA复杂系统实验室，庞培法布拉大学）；Universal Biology Institute, The University of Tokyo（东京大学全球生物研究所）；Department of Physics No. 1, Graduate School of Science, Kyoto University（京都大学理学研究科第一物理系）；Department of Physics, The University of Tokyo（东京大学物理学系）
Force-current structure in Markovian open quantum systems and its applications: geometric housekeeping-excess decomposition and thermodynamic trade-off relations
We report that population dynamics in fluctuating environment accompanies mathematically equivalent structure to steady state thermodynamics. By employing the structure, population growth in fluctuating environment is decomposed into housekeeping and excess parts. The housekeeping part represents the integral of stationary growth rate for each condition during a history of the environmental change. The excess part accounts for the excess growth generated when environment is switched. Focusing on the excess growth, we obtain Clausius inequality, which gives the upper bound of the excess growth.…
We propose a generalization of stochastic thermodynamics to systems of active particles, which move under the combined influence of stochastic internal self-propulsions (activity) and a heat bath. The main idea is to consider joint trajectories of particles' positions and self-propulsions. It is then possible to exploit formal similarity of an active system and a system consisting of two subsystems interacting with different heat reservoirs and coupled by a non-symmetric interaction. The resulting thermodynamic description closely follows the standard stochastic thermodynamics. In particular, …
We show that current fluctuations in a stochastic pump can be robustly mapped to fluctuations in a corresponding time-independent nonequilibrium steady state. We thus refine a recently proposed mapping so that it ensures equivalence of not only the averages, but also optimal representation of fluctuations in currents and density. Our mapping leads to a natural decomposition of the entropy production in stochastic pumps similar to the "housekeeping" heat. As a consequence of the decomposition of entropy production, the current fluctuations in weakly perturbed stochastic pumps are shown to satis…
Steady-state thermodynamics (SST) is a relatively newly emerging subfield of physics, which deals with transitions between steady states. In this paper, we find an SST-like structure in population dynamics of organisms that can sense their fluctuating environments. As heat is divided into two parts in SST, we decompose population growth into two parts: housekeeping growth and excess growth. Then, we derive the Clausius equality and inequality for excess growth. Using numerical simulations, we demonstrate how the Clausius inequality behaves depending on the magnitude of noise and strategies tha…
Thermo-Kinetic relations bound thermodynamic quantities such as entropy production with statistics of dynamical observables. We introduce a Thermo-Kinetic Relation to bound the entropy production or the non-adiabatic (Hatano-Sasa, excess) entropy production for overdamped Markov jump processes, possibly with time-varying rates and non stationary distributions. For stationary cases, this bound is akin to a Thermodynamic Uncertainty Relation, only involving absolute fluctuations rather than the mean square, thereby offering a better lower bound far from equilibrium. For non-stationary cases, thi…
A complete mathematical theory of nonequilibrium thermodynamics of stochastic systems in terms of master equations is presented. As generalizations of isothermal entropy and free energy, two functions of states play central roles: the Gibbs entropy $S$ and the relative entropy $F$, which are related via the stationary distribution of the stochastic dynamics. $S$ satisfies the fundamental entropy balance equation $dS/dt=e_p-h_d/T$ with entropy production rate $e_p\ge 0$ and heat dissipation rate $h_d$, while $dF/dt=-f_d\le 0$. For closed systems that satisfy detailed balance: $Te_p(t)=f_d(t)$. …
Glansdorff and Prigogine (1970) proposed a decomposition of the entropy production rate, which today is mostly known for Markov processes as the Hatano-Sasa approach. Their context was irreversible thermodynamics which, while ignoring fluctuations, still allows a somewhat broader treatment than the one based on the Master or Fokker-Planck equation. Glansdorff and Prigogine were the first to introduce a notion of excess entropy production rate $δ^2$EP and they suggested as sufficient stability criterion for a nonequilibrium macroscopic condition that $δ^2$EP be positive. We find for nonlinear d…
The meromorphic functional calculus developed in Part I overcomes the nondiagonalizability of linear operators that arises often in the temporal evolution of complex systems and is generic to the metadynamics of predicting their behavior. Using the resulting spectral decomposition, we derive closed-form expressions for correlation functions, finite-length Shannon entropy-rate approximates, asymptotic entropy rate, excess entropy, transient information, transient and asymptotic state uncertainty, and synchronization information of stochastic processes generated by finite-state hidden Markov mod…
