We investigate nonequilibrium underdamped Langevin dynamics of Brownian particles that interact through a harmonic potential with coupling constant $K$ and are in thermal contact with two heat baths at different temperatures. The system is characterized by a net heat flow and an entropy production in the steady state. We compare the entropy production of the harmonic system with that of Brownian particles linked with a rigid rod. The harmonic system may be expected to reduce to the rigid rod system in the infinite $K$ limit. However, we find that the harmonic system in the $K\to\infty$ limit produces more entropy than the rigid rod system. The harmonic system has the center of mass coordinate as a slow variable and the relative coordinate as a fast variable. By identifying the contributions of the degrees of freedom to the total entropy production, we show that the hidden entropy production by the fast variable is responsible for the extra entropy production. We discuss the $K$ dependence of each contribution.
In the general process of eliminating dynamic variables in Markovian models, there exists a difference in the irreversible entropy production between the original and reduced dynamics. We call this difference the hidden entropy production, since it is an invisible quantity when only the reduced system's view is provided. We show that this hidden entropy production obeys a new integral fluctuation theorem for the generic case where all variables are time-reversal invariant, therefore supporting the intuition that entropy production should decrease by coarse graining. It is found, however, that …
The transfer entropy is a well-established measure of information flow, which quantifies directed influence between two stochastic time series and has been shown to be useful in a variety fields of science. Here we introduce the transfer entropy of the backward time series called the backward transfer entropy, and show that the backward transfer entropy quantifies how far it is from dynamics to a hidden Markov model. Furthermore, we discuss physical interpretations of the backward transfer entropy in completely different settings of thermodynamics for information processing and the gambling wi…
The fluctuation theorem for entropy production is a remarkable symmetry of the distribution of produced entropy that holds universally in non-equilibrium steady states with Markovian dynamics. However, in systems with slow degrees of freedom that are hidden from the observer, it is not possible to infer the amount of produced entropy exactly. Previous work suggested that a relation similar to the fluctuation theorem may hold at least approximately for such systems if one considers an apparent entropy production. By extending the notion of apparent entropy production to discrete bipartite syste…
Recently, Kawaguchi and Nakayama (KN) [Phys. Rev. E {\bf 88}, 022147 (2013)] showed that the hidden entropy production associated with a coarse-graining procedure obeys the integral fluctuation theorem (IFT) if the original process does not involve any odd-parity variable that changes its sign under time reversal. This was interpreted as the evidence that odd-parity variables play an important role in the derivation of irreversible stochastic dynamics from time-reversible deterministic dynamics. In this Comment, we show that KN's approach is inadequate for describing the origin of irreversible…
Stochastic thermodynamics allows us to define heat and work for microscopic systems far from thermodynamic equilibrium, based on observations of their stochastic dynamics. However, a complete account of the energetics necessitates that all relevant nonequilibrium degrees of freedom are resolved, which is not feasible in many experimental situations. A simple approach is to map the visible dynamics onto a Markov model, which produces a lower-bound estimate of the entropy production. The bound, however, can be quite loose, especially when the visible dynamics only have small or vanishing observa…
We consider an open quantum system for which only a subset of all possible transitions are accessible, while the remaining ones are hidden from direct observation. Using a modification of the notion of quantum jump trajectories we introduce the coarse-grained hidden entropy, which quantifies the entropy production in the hidden subsystem conditioned on our observations of the visible part. The entropy production consisting of the sum of visible and coarse-grained hidden entropy is shown to satisfy an integral fluctuations theorem. Depending on the nature of the continuously occurring measureme…
In some situations in stochastic thermodynamics not all relevant slow degrees of freedom are accessible. Consequently, one adopts an effective description involving only the visible degrees of freedom. This gives rise to an apparent entropy production that violates standard fluctuation theorems. We present an analytically solvable model illustrating how the fluctuation theorems are modified. Furthermore, we define an alternative to the apparent entropy production: the marginal entropy production which fulfills the fluctuation theorems in the usual form. We show that the non-Markovianity of the…
Finding the entropy rate of Hidden Markov Processes is an active research topic, of both theoretical and practical importance. A recently used approach is studying the asymptotic behavior of the entropy rate in various regimes. In this paper we generalize and prove a previous conjecture relating the entropy rate to entropies of finite systems. Building on our new theorems, we establish series expansions for the entropy rate in two different regimes. We also study the radius of convergence of the two series expansions.
We revisit and analyze the thermodynamic efficiency of the Feynman-Smoluchowski (FS) ratchet, a classical thought experiment describing an autonomous heat-work converter. Starting from the full kinetics of the FS ratchet and deriving the exact forms of the hidden dissipations resulting from coarse-graining, we restate the historical controversy over its thermodynamic efficiency. The existence of hidden entropy productions implies that the standard framework of stochastic thermodynamics applied to the coarse-grained descriptions fails in capturing the dissipative feature of the system. In respo…
The entropy of a binary symmetric Hidden Markov Process is calculated as an expansion in the noise parameter epsilon. We map the problem onto a one-dimensional Ising model in a large field of random signs and calculate the expansion coefficients up to second order in epsilon. Using a conjecture we extend the calculation to 11th order and discuss the convergence of the resulting series.
Collections of self-propelled particles that move persistently by continuously consuming free energy are a paradigmatic example of active matter. In these systems, unlike Brownian "hot colloids", the breakdown of detailed balance yields a continuous production of entropy at steady state, even for an ideal active gas. We quantify the irreversibility for a non-interacting active particle in two dimensions by treating both conjugated and time-reversed dynamics. By starting with underdamped dynamics, we identify a hidden rate of entropy production required to maintain persistence and prevent the r…
We investigate a stationary process's crypticity---a measure of the difference between its hidden state information and its observed information---using the causal states of computational mechanics. Here, we motivate crypticity and cryptic order as physically meaningful quantities that monitor how hidden a hidden process is. This is done by recasting previous results on the convergence of block entropy and block-state entropy in a geometric setting, one that is more intuitive and that leads to a number of new results. For example, we connect crypticity to how an observer synchronizes to a proc…
Energy flow in bio-molecular motors and machines are vital to their function. Yet experimental observations are often limited to a small subset of variables that participate in energy transport and dissipation. Here we show, through a solvable Langevin model, that the seemingly hidden entropy production is measurable through the violation spectrum of the fluctuation-response relation of a slow observable. For general Markov systems with timescale separation, we prove that the violation spectrum exhibits a characteristic plateau in the intermediate frequency region. Despite its vanishing height…
Hidden Markov chains are widely applied statistical models of stochastic processes, from fundamental physics and chemistry to finance, health, and artificial intelligence. The hidden Markov processes they generate are notoriously complicated, however, even if the chain is finite state: no finite expression for their Shannon entropy rate exists, as the set of their predictive features is generically infinite. As such, to date one cannot make general statements about how random they are nor how structured. Here, we address the first part of this challenge by showing how to efficiently and accura…
We show that memory, feedback, and activity are all describable by the same unifying concept, that is non-reciprocal (NR) coupling. We demonstrate that characteristic thermodynamic features of these intrinsically nonequilibrium systems are reproduced by low-dimensional Markovian networks with NR coupling, which we establish as minimal models for such complex systems. NR coupling alone implies a violation of the fluctuation-dissipation relation, which is inevitably connected to entropy production, i.e., irreversibility. Hiding the NR coupled degrees of freedom renders non-Markovian, one-variabl…
The irreversible currents and entropy production rate of a dilute colloidal suspension are calculated using the linear irreversible thermodynamics and the linear response theory. The \anomalous" or \hidden" entropy recently discussed in the context of the stochastic thermodynamics is fully accounted in these classic frameworks. We show that the two distinct formulations lead to identical results as long as the local equilibrium assumption or, equivalently the linear response theory, is valid.
Systems with interacting degrees of freedom play a prominent role in stochastic thermodynamics. Our aim is to use the concept of detached path probabilities and detached entropy production for bipartite Markov processes and elaborate on a series of special cases including measurement-feedback systems, sensors and hidden Markov models. For these special cases we show that fluctuation theorems involving the detached entropy production recover known results which have been obtained separately before. Additionally, we show that the fluctuation relation for the detached entropy production can be us…
The thermodynamic uncertainty relation originally proven for systems driven into a non-equilibrium steady state (NESS) allows one to infer the total entropy production rate by observing any current in the system. This kind of inference scheme is especially useful when the system contains hidden degrees of freedom or hidden discrete states, which are not accessible to the experimentalist. A recent generalization of the thermodynamic uncertainty relation to arbitrary time-dependent driving allows one to infer entropy production not only by measuring current-observables but also by observing stat…
Entanglement exhibits universal behavior near the ground-state critical point where correlations are long-ranged and the thermodynamic entropy is vanishing. On the other hand, a quantum quench imparts extensive energy and results in a build-up of entropy, hence no critical behavior is expected at long times. In this work, we present a new paradigm in the quench dynamics of integrable spin chains which exhibit a ground-state order-disorder phase transition at a critical line. Specifically, we consider a quench along the critical line which displays a volume-law behavior of the entropy and expon…
Living systems operate far from thermal equilibrium by converting the chemical potential of ATP into mechanical work to achieve growth, replication or locomotion. Given time series observations of intra-, inter- or multicellular processes, a key challenge is to detect non-equilibrium behavior and quantify the rate of free energy consumption. Obtaining reliable bounds on energy consumption and entropy production directly from experimental data remains difficult in practice as many degrees of freedom typically are hidden to the observer, so that the accessible coarse-grained dynamics may not obv…
There is a simple bound on how fast the entanglement entropy of a subregion of a many-body quantum system can saturate in a quench: $t_\text{sat}\geq R/v_B$, where $t_\text{sat}$ is the saturation time, $R$ the radius of the largest inscribed sphere, and $v_B$ the butterfly velocity characterizing operator growth. By combining analytic and numerical approaches, we show that in systems with a holographic dual, the saturation time is equal to this lower bound for a variety of differently shaped entangling surfaces, implying that the dual black holes saturate the entanglement entropy as fast as p…
Loosely speaking, the Shannon entropy rate is used to gauge a stochastic process' intrinsic randomness; the statistical complexity gives the cost of predicting the process. We calculate, for the first time, the entropy rate and statistical complexity of stochastic processes generated by finite unifilar hidden semi-Markov models---memoryful, state-dependent versions of renewal processes. Calculating these quantities requires introducing novel mathematical objects (ε-machines of hidden semi-Markov processes) and new information-theoretic methods to stochastic processes.
