The entropy production rate (EPR) offers a quantitative measure of time reversal symmetry breaking in non-equilibrium systems. It can be defined either at particle level or at the level of coarse-grained fields such as density; the EPR for the latter quantifies the extent to which these coarse-grained fields behave irreversibly. In this work, we first develop a general method to compute the EPR of scalar Langevin field theories with additive noise. This large class of theories includes active versions of Model A (non-conserved density dynamics) and Model B (conserved) and also models where both types of dynamics are simultaneously present (such as Model AB). Treating the scalar field $ϕ$ (and its time derivative $\dotϕ$) as the sole observable(s), we arrive at an expression for the EPR that is non-negative for every field configuration and is quadratic in the time-antisymmetric component of the dynamics. Our general expression is a function of the quasipotential, which determines the full probability distribution for configurations, and is not generally calculable. To alleviate this difficulty, we present a small-noise expansion of the EPR, which only requires knowledge of the deterministic (mean-field) solution for the scalar field in steady state, which generally is calculable, at least numerically. We demonstrate this calculation for the case of Model AB. We then present a similar EPR calculation for Model AB with the conservative and non-conservative contributions to $\dotϕ= \dotϕ_{\rm A} + \dotϕ_{\rm B}$ viewed as separately observable quantities. The results are qualitatively different, confirming that the field-level EPR depends on the choice of coarse-grained information retained within the dynamical description.
Equilibrium is characterized by its fundamental properties such as the detailed balance, the fluctuation-dissipation relation, and no heat dissipation. Based on the stochastic thermodynamics, we show that these three properties are equivalent to each other in conventional Langevin thermal systems with microscopic reversibility. Thus, a conventional steady state has either all three properties (equilibrium) or none of them (nonequilibrium). In contrast, with velocity-dependent forces breaking the microscopic reversibility, we prove that the detailed balance and the fluctuation-dissipation relat…
This paper is the second in a series devoted to the study of Langevin systems subjected to a continuous time-delayed feedback control. The goal of our previous paper [Phys. Rev. E 91, 042114 (2015)] was to derive second-law-like inequalities that provide bounds to the average extracted work. Here we study stochastic fluctuations of time-integrated observables such as the heat exchanged with the environment, the extracted work, or the (apparent) entropy production. We use a path-integral formalism and focus on the long-time behavior in the stationary cooling regime, stressing the role of rare e…
We study the thermodynamic entropy as a Noether invariant in a stochastic process. Following the Onsager theory, we consider the Langevin equation for a thermodynamic variable in a thermally isolated system. By analyzing the Martin-Siggia-Rose-Janssen-de Dominicis action of the Langevin equation, we find that this action possesses a continuous symmetry in quasi-static processes, which leads to the thermodynamic entropy as the Noether invariant for the symmetry.
This paper shows an estimation of a lower bound on the total entropy production rate(EPR) for a system following Langevin dynamics with a single observed variable coarse-grained into a few discrete states, by invoking the underlying broken time-reversal symmetry. We estimate the irreversibility in waiting time distributions of the partially observed system and find the effect of spatial resolution on the mean dwell-time asymmetry and the EPR.
We study irreversible processes for nonlinear oscillators networks described by complex-valued Langevin equations that account for coupling to different thermo-chemical baths. Dissipation is introduced via non-Hermitian terms in the Hamiltonian of the model. We apply the stochastic thermodynamics formalism to compute explicit expressions for the entropy production rates. We discuss in particular the non-equilibrium steady states of the network characterised by a constant production rate of entropy and flows of energy and particle currents. For two specific examples, a one-dimensional chain and…
Generating an initial condition for a Langevin equation with memory is a non trivial issue. We introduce a generalisation of the Laplace transform as a useful tool for solving this problem, in which a limit procedure may send the extension of memory effects to arbitrary times in the past. This method allows us to compute average position, work, their variances and the entropy production rate of a particle dragged in a complex fluid by an harmonic potential, which could represent the effect of moving optical tweezers. For initial conditions in equilibrium we generalise the results by van Zon an…
Stochastic thermodynamics for delayed Langevin systems
讨论了时滞Langevin系统的随机热力学。利用ST的一般原理，可以用与无时滞系统类似的方法，很好地定义类第一定律的能量平衡和依赖于熵的熵s（t）。由于时间延迟的存在将额外的熵通量带入系统中，因此传统的第二定律$<{Δ{s_{tot}}}> \ge 0$不再成立，其中$Δ{s_{tot}}$表示沿着随机路径的总熵变化，并且$<...>$表示路径集合上的平均。借助于Fokker-Planck描述，我们引入了一个时延平均熵相关耗散泛函$η[{χ(t)}]$，它包含了时延平均力$\bar F({x,t})$沿路径$χ(t)$所做的功，等于无时延时的介质熵变$Δ{s_m}[ {x(t)}]$.我们证明了总耗散泛函R = Δs + η，其中$Δs$表示系统沿路径的熵变，服从$< R > \ge 0$，这可看作是时滞系统的第二定律.此外，还成功地验证了积分涨落定理$< <e^(-R)>=1 also holds true. We apply these concepts to a linear Langevin system with time delay and periodic external force. Numerical results demonstrate that the total entropy change $ < {Δ{ s_{tot} }} > $ co…
We derive a bound on generalized currents for Langevin systems in terms of the total entropy production in the system and its environment. For overdamped dynamics, any generalized current is bounded by the total rate of entropy production. We show that this entropic bound on the magnitude of generalized currents imposes power-efficiency tradeoff relations for ratchets in contact with a heat bath: Maximum efficiency---Carnot efficiency for a Smoluchowski-Feynman ratchet and unity for a flashing or rocking ratchet---can only be reached at vanishing power output. For underdamped dynamics, while t…
In this short note, we consider a system of two rotors, one of which interacts with a Langevin heat bath. We show that the system relaxes to its invariant measure (steady state) no faster than a stretched exponential $\exp(-c t^{1/2})$. This indicates that the exponent $1/2$ obtained earlier by the present authors and J.-P. Eckmann for short chains of rotors is optimal.
Using the recently developed covariant Ito-Langevin dynamics, we develop a non-equilibrium 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 non-equilibrium processes. We relate time-reversal…
Graham has shown in Z. Physik B 26, 397-405 (1977) that a fluctuation-dissipation relation can be imposed on a class of non-equilibrium Markovian Langevin equations that admit a stationary solution of the corresponding Fokker-Planck equation. The resulting equilibrium form of the Langevin equation is associated with a nonequilibrium Hamiltonian. Here we provide some explicit insight into how this Hamiltonian may loose its time reversal invariance and how the "reactive" and "dissipative" fluxes loose their distinct time reversal symmetries. The antisymmetric coupling matrix between forces and f…
We derive a universal bound on generalized currents in Langevin systems in terms of the mean-square fluctuations of the current and the total entropy production. This bound generalizes a relation previously found by Barato et al. to arbitrary times and transient states. Using the bound, we define a new efficiency for stochastic transport, which measures how close a given system comes to saturating the bound. The existence of such a bounded efficiency implies that stochastic transport is unavoidably accompanied by a fluctuations and dissipation, which cannot be reduced arbitrarily. We apply the…
Thermodynamic quantities of the hard-sphere system in the steady state with a small heat flux are calculated within the continuous media approach. Analytical expressions for pressure, internal energy, and entropy are found in the approximation of the fourth order in temperature gradients. It is shown that the gradient contributions to the internal energy depend on the volume, while the entropy satisfies the second law of thermodynamics for nonequilibrium processes. The calculations are performed for dimensions 3D, 2D, and 1D.
Modified fluctuation-dissipation and Einstein relation at non-equilibrium steady states
从G. S. Agarwal [Zeitschrift für Physik 252，25（1972）]，我们提出了一个统一的推导一些修改的波动耗散关系（MFDR），涉及到响应周围的小扰动非平衡稳态的相关性。使用这种形式主义，我们表明使用连续朗之万和离散主方程动力学导出的MFDR的速度形式的等价性。由此产生的添加剂修正爱因斯坦关系的例子使用闪烁棘轮模型的分子马达。
We derive bounds on the precision of fluctuating currents, which are valid for the steady state of underdamped Langevin dynamics. These bounds provide a generalization of the overdamped thermodynamic uncertainty relation to the finite-mass regime. In the overdamped case, the precision of a current is bounded by the entropy production. By contrast, the underdamped bound acquires two additional positive terms, which characterize the local mean acceleration and the fluctuations of the velocity. We generalize the bound to the cases of a magnetic field and anisotropic temperature, and derive a join…
We numerically determine the entropy for heat-conducting states, which is connected to the so-called excess heat considered as a basic quantity for steady-state thermodynamics in nonequilibrium. We adopt an efficient method to estimate the entropy from the bare heat current and find that the obtained entropy agrees with the familiar local equilibrium hypothesis well. Our method possesses a wider applicability than local equilibrium and opens a possibility to compare thermodynamic properties of complex systems with those in the local equilibrium. We further investigate the entropy for heat-cond…
We investigate entropy production in finitely slow transitions between nonequilibrium steady states in Markov jump processes using the improved adiabatic approximation method proposed by Takahashi, Fujii, Hino and Hayakawa [1]. This method provides systematic improvement of the adiabatic approximation on infinitely slow transitions from which we obtain nonadiabatic corrections and has a close relationship with the slow driving perturbation [2, 3]. We analyze two types of excess entropy production of Sagawa and Hayakawa [4] and Hatano and Sasa [5] as examples, and confirm that the leading adiab…
