Stochastic thermodynamics, fluctuation theorems and molecular machines

Collective Bistability of Pyridine-Furan Nanosprings Coupled by a Graphene Plate

Nanometer-sized molecular structures exhibiting mechanical-like switching between discrete states are of great interest for their potential uses in nanotechnology and materials science. Designing such structures and understanding how they can be combined to operate synchronously is a key to creating nanoscale functional units. Notable examples recently discovered using atomistic simulations are pyridine-furan and pyridine-pyrrole nanosprings. When slightly stretched in aqueous or organic solutions, these nanosprings exhibit bistable dynamics akin to Duffing nonlinear oscillators. Based on these findings, we designed a hybrid system consisting of several pyridine-furan nanosprings attached to a graphene plate in organic solvent and simulated the molecular dynamics of the construct. Our focus is on how the nanosprings coupled by a graphene plate work together, and whether such a design enables the nanosprings to respond synchronously to random perturbations and weak external stimuli. Molecular dynamics simulations of this specific construct are complemented by a theoretical model of coupled bistable systems to understand how the synchronization depends on coupling of bistable units.

Revealing Physical Mechanisms of Spatial Pattern Formation and Switching in Ecosystems via Nonequilibrium Landscape and Flux

Spatial patterns are widely observed in numerous nonequilibrium natural systems, often undergoing complex transitions and bifurcations, thereby exhibiting significant importance in many physical and biological systems such as embryonic development, ecosystem desertification, and turbulence. However, how spatial pattern formation emerges and how the spatial pattern switches are not fully understood. Here, a landscape-flux field theory is developed using the spatial mode expansion method to uncover the underlying physical mechanism of the pattern formation and switching. The landscape and flux field are identified as the driving force for spatial dynamics and applied this theory to the critical transitions between spatial vegetation patterns in semi-arid ecosystems, revealing that the nonequilibrium flux drives the switchings of spatial patterns. The emergence of pattern switching is revealed through the optimal pathways and how fast this occurs via the speed of pattern switching. Furthermore, both the averaged flux and the entropy production rate exhibit peaks near pattern switching boundaries, revealing dynamical and thermodynamical origins for pattern transitions, and further offering early warning signals for anticipating spatial pattern switching. This work thus reveals physical mechanisms on spatial pattern-switching in semi-arid ecosystems and, more generally, introduces a useful approach for quantifying spatial pattern switching in nonequilibrium systems, which further offers practical applications such as early warning signals for critical transitions of spatial patterns.

Immunological Avalanches in Renal Immune Diseases

The complex nature of immune system behavior in both autoimmune diseases and transplant rejection can be understood through the lens of avalanche dynamics in critical-point systems. This paper introduces the concept of the "immunological avalanche" as a framework for understanding unpredictable patterns of immune activity in both contexts. Just as avalanches represent sudden releases of accumulated potential energy, immune responses exhibit periods of apparent stability followed by explosive flares triggered by seemingly minor stimuli. The model presented here draws parallels between immune system behavior and other complex systems such as earthquakes, forest fires, and neuronal activity, where localized events can propagate into large-scale disruptions. In autoimmune conditions like systemic lupus erythematosus (SLE), which affects multiple organ systems including the kidneys in approximately 50% of patients, these dynamics manifest as alternating periods of remission and flares. Similarly, in transplant recipients, the immune system exhibits metastable behavior under constant allograft stimulation. This critical-point dynamics framework is characterized by threshold-dependent activation, positive feedback loops, and dynamic non-linearity. In autoimmune diseases, triggers such as UV light exposure, infections, or stress can initiate cascading immune responses. In transplant patients, longitudinal analysis reveals how monitoring oscillatory patterns in blood parameters and biological age markers can predict rejection risk. In a preliminary study on kidney transplant, all measured variables showed temporal instability. Proteinuria exhibited precise log-log linearity in power law analysis, confirming near-critical-point system behavior. Two distinct dynamic patterns emerged: large oscillations in eGFR, proteinuria, or biological age predicted declining function, while small oscillations indicated stability. During avalanche events, biological age increased dramatically, with partial reversal leaving persistent elevation after acute episodes. Understanding these dynamics has important implications for therapeutic approaches in both contexts. Key findings suggest that monitoring parameter oscillations, rather than absolute values, better indicates system instability and potential avalanche events. Additionally, biological age calculations provide valuable prognostic information, while proteinuria measurements offer efficient sampling for system dynamics assessment. This conceptual model provides a unifying framework for understanding the pathogenesis of both autoimmune and transplant-related immune responses, potentially leading to new perspectives in disease management and rejection prediction.

Time-Resolved Stochastic Dynamics of Quantum Thermal Machines

Steady-state quantum thermal machines are typically characterized by a continuous flow of heat between different reservoirs. However, at the level of discrete stochastic realizations, heat flow is unraveled as a series of abrupt quantum jumps, each representing an exchange of finite quanta with the environment. In this work, we present a framework that resolves the dynamics of quantum thermal machines into cycles classified as enginelike, coolinglike, or idle. We analyze the statistics of individual cycle types and their durations, enabling us to determine both the fraction of cycles useful for thermodynamic tasks and the average waiting time between cycles of a given type. Central to our analysis is the notion of intermittency, which captures the operational consistency of the machine by assessing the frequency and distribution of idle cycles. Our framework offers a novel approach to characterizing thermal machines, with significant relevance to experiments involving mesoscopic transport through quantum dots.

Empirical Study on Fluctuation Theorem for Volatility Cascade Processes in Stock Markets

This study investigates the properties of financial markets that arise from the multi-scale structure of volatility, particularly intermittency, by employing robust theoretical tools from nonequilibrium thermodynamics. Intermittency in velocity fields along spatial and temporal axes is a well-known phenomenon in developed turbulence, with extensive research dedicated to its structures and underlying mechanisms. In turbulence, such intermittency is explained through energy cascades, where energy injected at macroscopic scales is transferred to microscopic scales. Similarly, analogous cascade processes have been proposed to explain the intermittency observed in financial time series. In this work, we model volatility cascade processes in the stock market by applying the framework of stochastic thermodynamics to a Langevin system that describes the dynamics. We introduce thermodynamic concepts such as temperature, heat, work, and entropy into the analysis of financial markets. This framework allows for a detailed investigation of individual trajectories of volatility cascades across longer to shorter time scales. Further, we conduct an empirical study primarily using the normalized average of intraday logarithmic stock prices of the constituent stocks in the FTSE 100 Index listed on the London Stock Exchange (LSE), along with two additional data sets from the Tokyo Stock Exchange (TSE). Our Langevin-based model successfully reproduces the empirical distribution of volatility-defined as the absolute value of the wavelet coefficients across time scales-and the cascade trajectories satisfy the Integral Fluctuation Theorem associated with entropy production. A detailed analysis of the cascade trajectories reveals that, for the LSE data set, volatility cascades from larger to smaller time scales occur in a causal manner along the temporal axis, consistent with known stylized facts of financial time series. In contrast, for the two data sets from the TSE, while similar behavior is observed at smaller time scales, anti-causal behavior emerges at longer time scales.

Emergent short-range repulsion for attractively coupled active particles

We show that heterogeneity in self-propulsion speed can lead to the emergence of a robust effective short-range repulsion among active particles interacting via long-range attractive potentials. Using the example of harmonically coupled active Brownian particles, we analytically derive the stationary distribution of the pairwise distances and reveal that the heterogeneity in propulsion speeds induces a characteristic scale of repulsion between particles. This length scale algebraically increases with the difference in their self-propulsion speeds. In contrast to the conventional view that activity in active matter systems typically leads to effective attraction, our results demonstrate that activity can give rise to an emergent repulsive interaction. This phenomenon is universal, independent of the specific dynamics of the particles or the presence of thermal fluctuations. We also discuss possible experimental realization of this counter-intuitive phenomenon.

Information Gain Limit of Biomolecular Computation

Biomolecules stochastically occupy different configurations that correspond to distinct functional states. Changing biochemical inputs such as rate constants alters the output probability distribution of configurations, and thus constitutes a form of computation. In the cell, such computations are often coupled to thermodynamic forces such as ATP hydrolysis that drive systems far from equilibrium, resulting in energy expenditure even during times when computations are not being performed. The information-theoretic advantage of this costly computational paradigm is unclear. Here we introduce a theoretical framework showing how much the thermodynamic force enables changes in probability distributions, quantified by the information gain, beyond what is possible at equilibrium. Using this framework, we derive a general expression relating the force to the maximum information gain in an arbitrary computation, revealing how small input changes can exponentially alter outputs. We numerically show that biomolecular systems can closely approach this universal bound, illustrating how energy expenditure is needed to achieve the information processing capabilities observed in nature.

Spontaneous generation of angular momentum in chiral active crystals

We study a two-dimensional chiral active crystal composed of underdamped chiral active particles. These particles, characterized by intrinsic handedness and persistence, interact via linear forces derived from harmonic potentials. Chirality plays a pivotal role in shaping the system's behavior: it reduces displacement and velocity fluctuations while inducing cross-spatial correlations among different Cartesian components of velocity. These features distinguish chiral crystals from their non-chiral counterparts, leading to the emergence of net angular momentum, as predicted analytically. This angular momentum, driven by the torque generated by the chiral active force, exhibits a non-monotonic dependence on the degree of chirality. Additionally, it contributes to the entropy production rate, as revealed through a path-integral analysis. We investigate the dynamic properties of the crystal in both Fourier and real space. Chirality induces a non-dispersive peak in the displacement spectrum, which underlies the generation of angular momentum and oscillations in time-dependent autocorrelation functions or mean-square displacement, all of which are analytically predicted.

Flow of Energy and Information in Molecular Machines

Molecular machines transduce free energy between different forms throughout all living organisms. Unlike their macroscopic counterparts, molecular machines are characterized by stochastic fluctuations, overdamped dynamics, and soft components, and operate far from thermodynamic equilibrium. In addition, information is a relevant free energy resource for molecular machines, leading to new modes of operation for nanoscale engines. Toward the objective of engineering synthetic nanomachines, an important goal is to understand how molecular machines transduce free energy to perform their functions in biological systems. In this review, we discuss the nonequilibrium thermodynamics of free energy transduction within molecular machines, with a focus on quantifying energy and information flows between their components. We review results from theory, modeling, and inference from experiments that shed light on the internal thermodynamics of molecular machines, and ultimately explore what we can learn from considering these interactions.

Transient Thermal Energy Harvesting at a Single Temperature Using Nonlinearity

The authors present an in-depth theoretical study of two nonlinear circuits capable of transient thermal energy harvesting at one temperature. The first circuit has a storage capacitor and diode connected in series. The second circuit has three storage capacitors, and two diodes arranged for full wave rectification. The authors solve both Ito-Langevin and Fokker-Planck equations for both circuits using a large parameter space including capacitance values and diode quality. Surprisingly, using diodes one can harvest thermal energy at a single temperature by charging capacitors. However, this is a transient phenomenon. In equilibrium, the capacitor charge is zero, and this solution alone satisfies the second law of thermodynamics. The authors found that higher quality diodes provide more stored charge and longer lifetimes. Harvesting thermal energy from the ambient environment using diode nonlinearity requires capacitors to be charged but then disconnected from the circuit before they have time to discharge.

Electrostatic All-Passive Force Clamping of Charged Nanoparticles

In the past decades, many techniques have been explored for trapping microscopic and nanoscopic objects, but the investigation of nano-objects under arbitrary forces and conditions remains nontrivial. One fundamental case concerns the motion of a particle under a constant force, known as force clamping. Here, we employ metallic nanoribbons embedded in a glass substrate in a capacitor configuration to generate a constant electric field on a charged nanoparticle in a water-filled glass nanochannel. We estimate the force fields from Brownian trajectories over several micrometers and confirm the constant behavior of the forces both numerically and experimentally. Furthermore, we manipulate the diffusion and relaxation times of the nanoparticles by tuning the charge density on the electrode. Our highly compact and controllable setting allows for the trapping and force-clamping of charged nanoparticles in a solution, providing a platform for investigating nanoscopic diffusion phenomena.

Multilevel irreversibility reveals higher-order organization of nonequilibrium interactions in human brain dynamics

Information processing in the human brain can be modeled as a complex dynamical system operating out of equilibrium with multiple regions interacting nonlinearly. Yet, despite extensive study of the global level of nonequilibrium in the brain, quantifying the irreversibility of interactions among brain regions at multiple levels remains an unresolved challenge. Here, we present the Directed Multiplex Visibility Graph Irreversibility framework, a method for analyzing neural recordings using network analysis of time-series. Our approach constructs directed multilayer graphs from multivariate time-series where information about irreversibility can be decoded from the marginal degree distributions across the layers, which each represents a variable. This framework is able to quantify the irreversibility of every interaction in the complex system. Applying the method to magnetoencephalography recordings during a long-term memory recognition task, we quantify the multivariate irreversibility of interactions between brain regions and identify the combinations of regions which showed higher levels of nonequilibrium in their interactions. For individual regions, we find higher irreversibility in cognitive versus sensorial brain regions while for pairs, strong relationships are uncovered between cognitive and sensorial pairs in the same hemisphere. For triplets and quadruplets, the most nonequilibrium interactions are between cognitive-sensorial pairs alongside medial regions. Combining these results, we show that multilevel irreversibility offers unique insights into the higher-order, hierarchical organization of neural dynamics from the perspective of brain network dynamics.

Perspective: Time irreversibility in systems observed at coarse resolution

A broken time-reversal symmetry, i.e., broken detailed balance, is central to non-equilibrium physics and is a prerequisite for life. However, it turns out to be quite challenging to unambiguously define and quantify time-reversal symmetry (and violations thereof) in practice, that is, from observations. Measurements on complex systems have a finite resolution and generally probe low-dimensional projections of the underlying dynamics, which are well known to introduce memory. In situations where many microscopic states become "lumped" onto the same observable "state" or when introducing "reaction coordinates" to reduce the dimensionality of data, signatures of a broken time-reversal symmetry in the microscopic dynamics become distorted or masked. In this Perspective, we highlight why, in defining and discussing time-reversal symmetry and quantifying its violations, the precise underlying assumptions on the microscopic dynamics, the coarse graining, and further reductions are not a technical detail. These assumptions decide whether the conclusions that are drawn are physically sound or inconsistent. We summarize recent findings in the field and reflect upon key challenges.

Physics-informed deep learning for stochastic particle dynamics estimation

Single-particle tracking has enabled quantitative studies of complex systems, providing nanometer localization precision and millisecond temporal resolution in heterogeneous environments. However, at micro- or nanometer scales, probe dynamics become inherently stochastic due to Brownian motion and complex interactions, leading to varied diffusion behaviors. Typically, analysis of such trajectory data involves certain moving-window operation and assumes the existence of some pseudo-steady states, particularly when evaluating predefined parameters or specific types of diffusion modes. Here, we introduce the stochastic particle-informed neural network (SPINN), a physics-informed deep learning framework that integrates stochastic differential equations to model and infer particle diffusion dynamics. The SPINN autonomously explores parameter spaces and distinguishes between deterministic and stochastic components with single-frame resolution. Using the anomalous diffusion dataset, we validated SPINN's ability to reduce frame-to-frame variability while preserving key statistical correlations, allowing for accurate characterization of different stochastic processes. When applied to the diffusion of single gold nanorods in hydrogels, the SPINN revealed enhanced microrheological properties during hydrogel gelation and uncovered interfacial dynamics during dextran/tetra-PEG liquid-liquid phase separation. By improving the temporal resolution of stochastic dynamics, the SPINN facilitates the estimation and prediction of complex diffusion behaviors, offering insights into underlying physical mechanisms at mesoscopic scales.

Metareview: a survey of active matter reviews

In the past years, the amount of research on active matter has grown extremely rapidly, a fact that is reflected in particular by the existence of more than 1000 reviews on this topic. Moreover, the field has become very diverse, ranging from theoretical studies of the statistical mechanics of active particles to applied work on medical applications of microrobots and from biological systems to artificial swimmers. This makes it very difficult to get an overview over the field as a whole. Here, we provide such an overview in the form of a metareview article that surveys the existing review articles and books on active matter. Thereby, this article provides a useful starting point for finding literature about a specific topic.

Stochastic thermodynamic bounds on logical circuit operation

Using a thermodynamically consistent, mesoscopic model for modern complementary metal-oxide-semiconductor transistors, we study an array of logical circuits and explore how their function is constrained by recent thermodynamic uncertainty relations when operating near thermal energies. For a single NOT gate, we find operating direction-dependent dynamics and a trade-off between dissipated heat and operation time certainty. For a memory storage device, we find an exponential relationship between the memory retention time and energy required to sustain that memory state. For a clock, we find that the certainty in the cycle time is maximized at biasing voltages near thermal energy, as is the trade-off between this certainty and the heat dissipated per cycle. We identify a control mechanism that can increase the cycle time certainty without an offsetting increase in heat dissipation by working at a resonance condition for the clock. These results provide a framework for assessing the thermodynamic costs of realistic computing devices, allowing for circuits to be designed and controlled for thermodynamically optimal operation.

Dynamical phase transitions in the nonreciprocal Ising model

Nonreciprocal interactions in many-body systems lead to time-dependent states, commonly observed in biological, chemical, and ecological systems. The stability of these states in the thermodynamic limit and the critical behavior of the phase transition from static to time-dependent states are not yet fully understood. To address these questions, we study a minimalistic system endowed with nonreciprocal interactions: an Ising model with two spin species having opposing goals. The mean-field equation predicts three stable phases: disorder, static order, and a time-dependent swap phase. Large-scale numerical simulations support the following: (i) in two dimensions, the swap phase is destabilized by defects; (ii) in three dimensions, the swap phase is stable and has the properties of a time crystal; (iii) the transition from disorder to swap in three dimensions is characterized by the critical exponents of the 3D XY model and corresponds to the breaking of a continuous symmetry, time translation invariance; (iv) when the two species have fully antisymmetric couplings, the static-order phase is unstable in any finite dimension due to droplet growth; and (v) in the general case of asymmetric couplings, static order can be restored by a droplet-capture mechanism preventing the droplets from growing indefinitely. We provide details on the full phase diagram, which includes first- and second-order-like phase transitions, and study how the system coarsens into swap and static-order states.

Irreversibility and Energy Transfer at Non-MHD Scales in a Magnetospheric Current Disruption Event

Irreversibility and the processes occurring at ion and sub-ion scales are key challenges in understanding energy dissipation in non-collisional space plasmas. Recent advances have significantly improved the characterization of irreversibility and energy transfer across scales in turbulent fluid-like media, using high-order correlation functions and testing the validity of certain fluctuation relations. In this study, we explore irreversibility at non-MHD scales during a magnetospheric current disruption event. Our approach involves analyzing the asymmetric correlation function, assessing the validity of a fluctuation relation, and investigating delayed coupling between different scales to reveal evidence of a cascading mechanism. The results clearly demonstrate the irreversible nature of fluctuations at ion and sub-ion scales. Additionally, we provide potential evidence for an energy cascading mechanism occurring over short time delays.

Decomposition of metric tensor in thermodynamic geometry in terms of relaxation timescales

Geometrical methods are extensively applied to thermodynamics, including stochastic thermodynamics. In the case of a slow-driving linear response regime, a geometrical framework, known as thermodynamic geometry, is established. The key to this framework is the thermodynamic length characterized by a metric tensor defined in the space of controlling variables. As the metric tensor is given in terms of the equilibrium time-correlation functions of the thermodynamic forces, it contains the information on timescales, which may be useful for analyzing the performance of heat engines. In this paper, we show that the metric tensor for underdamped Langevin dynamics can be decomposed in terms of the relaxation times of a system itself, which govern the timescales of the equilibrium time-correlation functions of the thermodynamic forces. As an application of the decomposition of the metric tensor, we demonstrate that it is possible to achieve Carnot efficiency at finite power by taking the vanishing limit of relaxation times without breaking trade-off relations between efficiency and power of heat engines in terms of thermodynamic geometry.

Nonequilibrium Relaxation Inequality on Short Timescales

An integral relation is derived from the Fokker-Planck equation which connects the steady-state probability currents with the dynamics of relaxation on short timescales in the limit of small perturbation fields. As a consequence of this integral relation, a general lower bound on the steady-state entropy production is obtained. Two particular ensembles of perturbation fields are then considered, respectively constant gradients and density displacements, and correspondingly two different averaging-based thermodynamic bounds are derived from the integral relation. These provide feasible methods to estimate the steady-state entropy production from relaxation experiments.

Three Optima of Thermoelectric Conversion: Insights from the Constant Property Model

Starting from Ioffe's description of a thermoelectric converter, we recover the optimal working points of conversion: the point of maximum efficiency and the one of maximal power. Inspired by biological converters' optimization, we compute a third optimal point associated with cost of energy (COE). This alternative cost function corresponds to the amount of heat exchanged with the cold reservoir per unit of electric current used. This work emphasizes the symmetry between the efficiency and performance coefficient of the electric generator and heat pump modes. It also reveals the relation between their optimal working points.

Precisely controlled colloids: a playground for path-wise non-equilibrium physics

We investigate path-wise observables in experiments on driven colloids in a periodic light field to dissect selected intricate transport features, kinetics, and transition-path time statistics out of thermodynamic equilibrium. These observables directly reflect the properties of individual paths in contrast to the properties of an ensemble of particles, such as radial distribution functions or mean-squared displacements. In particular, we present two distinct albeit equivalent formulations of the underlying stochastic equation of motion, highlight their respective practical relevance, and show how to interchange between them. We discuss conceptually different notions of local velocities and interrogate one- and two-sided first-passage and transition-path time statistics in and out of equilibrium. Our results reiterate how path-wise observables may be employed to systematically assess the quality of experimental data and demonstrate that, given sufficient control and sampling, one may quantitatively verify subtle theoretical predictions.

On the Numerical Integration of the Fokker-Planck Equation Driven by a Mechanical Force and the Bismut-Elworthy-Li Formula

Optimal control theory aims to find an optimal protocol to steer a system between assigned boundary conditions while minimizing a given cost functional in finite time. Equations arising from these types of problems are often non-linear and difficult to solve numerically. In this article, we describe numerical methods of integration for two partial differential equations that commonly arise in optimal control theory: the Fokker-Planck equation driven by a mechanical potential for which we use the Girsanov theorem; and the Hamilton-Jacobi-Bellman, or dynamic programming, equation for which we find the gradient of its solution using the Bismut-Elworthy-Li formula. The computation of the gradient is necessary to specify the optimal protocol. Finally, we give an example application of the numerical techniques to solving an optimal control problem without spacial discretization using machine learning.

Heat as a Witness of Quantum Properties

We present a new approach for witnessing quantum resources, such as entanglement and coherence, based on heat generation. Inspired by Maxwell's demon, we ask what the optimal heat exchange between a quantum system and a thermal environment is when the process is assisted by a quantum memory. We derive fundamental energy constraints in this scenario and show that quantum states can reveal nonclassical signatures via heat exchange. This approach leads to a heat-based witness for quantum properties, offering an alternative to system-specific measurements, as it only relies on fixed energy measurements in a thermal ancilla. We illustrate our findings with the detection of entanglement in isotropic states and coherence in two-spin systems interacting with a single-mode electromagnetic field.

Quantum Detailed Fluctuation Theorem in Curved Spacetimes: The Observer Dependent Nature of Entropy Production

The interplay between thermodynamics, general relativity, and quantum mechanics has long intrigued researchers. Recently, important advances have been obtained in thermodynamics, mainly regarding its application to the quantum domain through fluctuation theorems. In this Letter, we apply Fermi normal coordinates to report a fully general relativistic detailed quantum fluctuation theorem based on the two point measurement scheme. We demonstrate how the spacetime curvature can produce entropy in a localized quantum system moving in a general spacetime. The example of a quantum harmonic oscillator living in an expanding universe is presented. This result implies that entropy production is strongly observer dependent and deeply connects the arrow of time with the causal structure of the spacetime.

Coarse-Graining Chemical Networks by Trimming to Preserve Energy Dissipation

Continuous production of entropy and the corresponding energy dissipation is a defining characteristic of nonequilibrium systems. When a system's full chemical kinetic description is known, its entropy production rate can be computed from the microscopic rate constants. However, such a calculation typically underestimates energy dissipation when the states of the underlying system are mesoscopic, i.e., when they combine multiple microscopic states, a situation typical in experimental measurements with finite resolution. It is unknown whether there is a mesoscopic coarse-graining procedure that produces fewer states but allows for precise entropy production calculations. Here we develop a universal coarse-graining procedure that we call "trimming", in which microscopic states of the original Markov network are progressively eliminated but the fluxes between remaining states are exactly preserved. We demonstrate that this procedure also preserves entropy production as long as no dissipative loops are eliminated. We apply our method to several examples illustrating how trimming affects local network topology.

Bridging Freidlin-Wentzell large deviations theory and stochastic thermodynamics

For overdamped Langevin systems subjected to weak thermal noise and nonconservative forces, we establish a connection between Freidlin-Wentzell large deviations theory and stochastic thermodynamics. First, we derive a series expansion of the quasipotential around the detailed-balance solution, that is, the system's free energy, and identify the conditions for the linear response regime to hold, even far from equilibrium. Second, we prove that the escape rate from dissipative fixed points of the macroscopic dynamics is bounded by the entropy production of trajectories that relax into and escape from the attractors. These results provide the foundation to study the nonequilibrium thermodynamics of dissipative metastable states.

Power-efficiency constraint for chemical motors

Chemical gradients provide the primordial energy for biological functions by driving the mechanical movement of microscopic engines. Their thermodynamic properties remain elusive, especially concerning the dynamic change in energy demand in biological systems. In this article, we derive a constraint relation between the output power and the conversion efficiency for a chemically fueled steady-state rotary motor analogous to the F_{0} motor of ATPase. We find that the efficiency at maximum power is half of the maximum quasistatic efficiency. These findings shall aid in the understanding of natural chemical engines and inspire the manual design and control of chemically fueled microscale engines.

Second-law violating events are not rare in nonequilibrium nonsteady states

For nonequilibrium systems in which fluctuations are important, it is well established that there can be events that violate the second law of thermodynamics on the trajectory level. For nonequilibrium steady-states, these second-law violating events are rare, but this may not be true for far-from-equilibrium nonsteady processes. In the paradigm system of a Brownian particle trapped under a time-dependent compressing potential, we demonstrate that second-law violating trajectories can outnumber the second-law obeying ones. In particular, for abrupt compressing harmonic and anharmonic potentials, analytic expressions for the total entropy production distribution and the fraction of second-law violating events are derived to show explicitly that second-law violating events can be of significant majority. These results are further confirmed in experiments.

Uncovering dissipation from coarse observables: A case study of a random walk with unobserved internal states

Inferring underlying microscopic dynamics from low-dimensional experimental signals is a central problem in physics, chemistry, and biology. As a trade-off between molecular complexity and the low-dimensional nature of experimental data, mesoscopic descriptions such as the Markovian master equation are commonly used. The states in such descriptions usually include multiple microscopic states, and the ensuing coarse-grained dynamics are generally non-Markovian. It is frequently assumed that such dynamics can nevertheless be described as a Markov process because of the timescale separation between slow transitions from one observed coarse state to another and the fast interconversion within such states. Here, we use a simple model of a molecular motor with unobserved internal states to highlight that (1) dissipation estimated from the observed coarse dynamics may significantly underestimate microscopic dissipation even in the presence of timescale separation and even when mesoscopic states do not contain dissipative cycles and (2) timescale separation is not necessarily required for the Markov approximation to give the exact entropy production, provided that certain constraints on the microscopic rates are satisfied. When the Markov approximation is inadequate, we discuss whether including memory effects can improve the estimate. Surprisingly, when we do so in a "model-free" way by computing the Kullback-Leibler divergence between the observed probability distributions of forward trajectories and their time reverses, this leads to poorer estimates of entropy production. Finally, we argue that alternative approaches, such as hidden Markov models, may uncover the dissipative nature of the microscopic dynamics even when the observed coarse trajectories are completely time-reversible.

Maximizing Free Energy Gain

Maximizing the amount of work harvested from an environment is important for a wide variety of biological and technological processes, from energy-harvesting processes such as photosynthesis to energy storage systems such as fuels and batteries. Here, we consider the maximization of free energy-and by extension, the maximum extractable work-that can be gained by a classical or quantum system that undergoes driving by its environment. We consider how the free energy gain depends on the initial state of the system while also accounting for the cost of preparing the system. We provide simple necessary and sufficient conditions for increasing the gain of free energy by varying the initial state. We also derive simple formulae that relate the free energy gained using the optimal initial state rather than another suboptimal initial state. Finally, we demonstrate that the problem of finding the optimal initial state may have two distinct regimes, one easy and one difficult, depending on the temperatures used for preparation and work extraction. We illustrate our results on a simple model of an information engine.

Maximum-Power Stirling-like Heat Engine with a Harmonically Confined Brownian Particle

Heat engines transform thermal energy into useful work, operating in a cyclic manner. For centuries, they have played a key role in industrial and technological development. Historically, only gases and liquids have been used as working substances, but the technical advances achieved in recent decades allow for expanding the experimental possibilities and designing engines operating with a single particle. In this case, the system of interest cannot be addressed at a macroscopic level and their study is framed in the field of stochastic thermodynamics. In the present work, we study mesoscopic heat engines built with a Brownian particle submitted to harmonic confinement and immersed in a fluid acting as a thermal bath. We design a Stirling-like heat engine, composed of two isothermal and two isochoric branches, by controlling both the stiffness of the harmonic trap and the temperature of the bath. Specifically, we focus on the irreversible, non-quasi-static case-whose finite duration enables the engine to deliver a non-zero output power. This is a crucial aspect, which enables the optimisation of the thermodynamic cycle by maximising the delivered power-thereby addressing a key goal at the practical level. The optimal driving protocols are obtained by using both variational calculus and optimal control theory tools. Furthermore, we numerically explore the dependence of the maximum output power and the corresponding efficiency on the system parameters.

Stochastic Thermodynamics of a Linear Optical Cavity Driven on Resonance

We present a complete framework of stochastic thermodynamics for a single-mode linear optical cavity driven on resonance. We first show that the steady-state intracavity field follows the equilibrium Boltzmann distribution. The effective temperature is given by the noise variance, and the equilibration rate is the dissipation rate. Next, we derive expressions for internal energy, work, heat, and free energy of light in a cavity and formulate the first and second laws of thermodynamics for this system. We then analyze fluctuations in work and heat and show that they obey universal statistical relations known as fluctuation theorems. Finite time corrections to the fluctuation theorems are also discussed. Additionally, we show that work fluctuations obey Crooks' fluctuation theorem which is a paradigm for understanding emergent phenomena and estimating free energy differences. The significance of our results is twofold. On one hand, our work positions optical cavities as a unique platform for fundamental studies of stochastic thermodynamics. On the other hand, our work paves the way for improving the energy efficiency and information processing capabilities of laser-driven optical resonators using a thermodynamics based prescription.

Stochastic thermodynamics of Fisher information

In this manuscript, we investigate the stochastic thermodynamics of Fisher information (FI), meaning we characterize both the fluctuations of FI, introducing a statistics of that quantity, and thermodynamic quantities. We introduce two initial conditions: an equilibrium initial condition and a minimum entropy initial condition, both under a protocol that drives the system to equilibrium. Its results indicate a dependence of the average FI on both the initial condition and path taken. Furthermore, the results indicate that the chosen parameter directly affects the FI of thermodynamic quantities, such as irreversible work and entropy, along with fluctuations of a stochastic FI. Last, we assess the further role of FI of the distribution of thermal quantities within the context of thermostatistical inequalities.

Trade-off between coherence and dissipation for excitable phase oscillators

Thermodynamic uncertainty relation (TUR) bounds coherence in stochastic oscillatory systems. In this paper, we show that both dynamical and thermodynamic bounds play important roles for the excitable oscillators, e.g., neurons. Firstly, we investigate the trade-off between coherence and dissipation both in the sub- and superthreshold regions for a single excitable unit, where both the TUR and the saddle-node on an invariant circle (SNIC) bounds constrain the fluctuation of interspike intervals. Secondly, we show that the widely studied phenomenon called coherence resonance, where there exists a noise strength to make the oscillatory responses of the system most coherent, is also bounded by the TUR in the one-dimensional excitable phase model. Finally, we study the coherence-dissipation relation in ensembles of strongly coupled excitable oscillators.

Information geometry approach to quantum stochastic thermodynamics

Recent advancements have revealed new links between information geometry and classical stochastic thermodynamics, particularly through the Fisher information (FI) with respect to time. Recognizing the nonuniqueness of the quantum Fisher metric in Hilbert space, we exploit the fact that any quantum Fisher information (QFI) can be decomposed into a metric-independent incoherent part and a metric-dependent coherent contribution. We demonstrate that the incoherent component of any QFI can be directly linked to entropic acceleration, and for GKSL dynamics with local detailed balance, to the rate of change of generalized thermodynamic forces and entropic flow, paralleling the classical results. Furthermore, we tighten a classical uncertainty relation between the geometric uncertainty of a path in state space and the time-averaged rate of information change and demonstrate that it also holds for quantum systems. We generalize a classical geometric bound on the entropy rate for far-from-equilibrium processes by incorporating a nonnegative quantum contribution that arises from the geometric action due to coherent dynamics. Finally, we apply an information-geometric analysis to the recently proposed quantum-thermodynamic Mpemba effect, demonstrating this framework's ability to capture thermodynamic phenomena.

Nonequilbrium physics of generative diffusion models

Generative diffusion models apply the concept of Langevin dynamics in physics to machine learning, attracting a lot of interest from engineering, statistics, and physics, but a complete picture of inherent mechanisms is still lacking. In this paper, we provide a transparent physics analysis of diffusion models, formulating the fluctuation theorem, entropy production, equilibrium measure, and Franz-Parisi potential to understand the dynamic process and intrinsic phase transitions. Our analysis is rooted in a path integral representation of both forward and backward dynamics, and in treating the reverse diffusion generative process as a statistical inference, where the time-dependent state variables serve as a quenched disorder akin to that in spin glass theory. Our study thus links stochastic thermodynamics, statistical inference and geometry-based analysis together to yield a coherent picture of how the generative diffusion models work.

Dynamics of microscale and nanoscale systems in the weak-memory regime: A mathematical framework beyond the Markov approximation

The visible dynamics of small-scale systems are strongly affected by unobservable degrees of freedom, which can belong to either external environments or internal subsystems and almost inevitably induce memory effects. Formally, such inaccessible degrees of freedom can be systematically eliminated from essentially any microscopic model through projection operator techniques, which result in nonlocal time evolution equations. This article investigates how and under what conditions locality in time can be rigorously restored beyond the standard Markov approximation, which generally requires the characteristic timescales of accessible and inaccessible degrees of freedom to be sharply separated. Specifically, we consider nonlocal time evolution equations that are autonomous and linear in the variables of interest. For this class of models, we prove a mathematical theorem that establishes a well-defined weak-memory regime, where faithful local approximations exist, even if the relevant timescales are of comparable order of magnitude. The generators of these local approximations, which become exact in the long-time limit, are time independent and can be determined to arbitrary accuracy through a convergent perturbation theory in the memory strength, where the Markov generator is recovered in first order. For illustration, we work out three simple, yet instructive, examples covering coarse-grained Markov jump networks, semi-Markov jump processes, and generalized Langevin equations.

Foundations of algorithmic thermodynamics

Gács's coarse-grained algorithmic entropy leverages universal computation to quantify the information content of any given physical state. Unlike the Boltzmann and Gibbs-Shannon entropies, it requires no prior commitment to macrovariables or probabilistic ensembles, rendering it applicable to settings arbitrarily far from equilibrium. For measure-preserving dynamical systems equipped with a Markovian coarse graining, we prove a number of fluctuation inequalities. These include algorithmic versions of Jarzynski's equality, Landauer's principle, and the second law of thermodynamics. In general, the algorithmic entropy determines a system's actual capacity to do work from an individual state, whereas the Gibbs-Shannon entropy gives only the mean capacity to do work from a state ensemble that is known a priori.

Phase coexistence in a weakly stochastic reaction-diffusion system

We investigate phase coexistence in a weakly stochastic reaction-diffusion system without assuming a continuum description. Concretely, for (2N+1) diffusion-coupled vessels in which a chemical reaction exhibiting bistability occurs, we derive a condition for the phase coexistence in the limit N→∞. We then find that the phase coexistence condition depends on the rate of hopping between neighboring vessels. The conditions in the high- and low-hopping-rate limits are expressed in terms of two different potentials which are determined from the chemical reaction model in a single vessel.

Deviation of a system of nonreciprocally coupled harmonic oscillators from a conservative system

Discrete systems of coupled linear mechanical oscillators with nonreciprocal interaction are a model for a variety of physical systems. In general, the presence of nonreciprocal interactions renders their dynamics nonconservative, but under certain conditions it remains conservative. In this paper we show which thermodynamic properties induced by nonreciprocity can be observed in conservative systems and which are specific to nonconservative systems. To this end, we formulate a criterion for identifying conservative systems and construct a measure to quantify the deviation from conservativity.

Fluctuation-response relation in nonequilibrium systems and active matter

We use dynamic equations to derive a relation between correlation functions and response or relaxation functions in many-body systems. The relation is very general and holds both in equilibrium, when the usual fluctuation-dissipation theorem is valid, and to linear order in an expansion about arbitrary nonequilibrium states, when it is not. It provides a simple way to observe microscopic correlations via a system's macroscopic response, even in situations where the correlations are difficult to observe directly, and the response is not determined by the standard commutator (i.e., antisymmetrized) correlation functions. We illustrate our results by discussing fluids, both in equilibrium and nonequilibrium states, as well as several active-matter systems.

Ergodicity breaking and restoration in models of heat transport with microscopic reversibility

The behavior of lattice models in which time reversibility is enforced at the level of trajectories (microscopic reversibility) is studied analytically. Conditions for ergodicity breaking are explored, and a few examples of systems characterized by an additional conserved quantity besides energy are presented. All the systems are characterized by ergodicity restoration when put in contact with a thermal bath, except for specific choices of the interactions between the atoms in the system and the bath. The study shows that the additional conserved quantities return to play a role in nonequilibrium conditions. The similarities with the behavior of some mesoscale systems, in which the transition rates satisfy detailed balance but not microscopic reversibility, are discussed.

Second Law of Thermodynamics without Einstein Relation

Materials that are constantly driven out of thermodynamic equilibrium, such as active and living systems, typically violate the Einstein relation. This may arise from active contributions to particle fluctuations which are unrelated to the dissipative resistance of the surrounding medium. We show that in these cases the widely used relation between informatic entropy production and heat dissipation does not hold. Consequently, fluctuation relations for the mechanical work, such as the Jarzynski and Crooks theorems, are invalid. We relate the breaking of the correspondence between entropy production and heat dissipation to departure from the fluctuation-dissipation theorem. We propose a temperaturelike variable that restores this correspondence and gives rise to a generalized second law of thermodynamics, whereby the dissipated heat is necessarily non-negative and vanishes at equilibrium. The Clausius inequality, Carnot maximum efficiency theorem, and relation between the extractable work and the change of free energy are recovered as well.

Heat Transport Hysteresis Generated Through Frequency Switching of a Time-Dependent Temperature Gradient

A stochastic energetics framework is applied to examine how periodically shifting the frequency of a time-dependent oscillating temperature gradient affects heat transport in a nanoscale molecular model. We specifically examine the effects that frequency switching, i.e., instantaneously changing the oscillation frequency of the temperature gradient, has on the shape of the heat transport hysteresis curves generated by a particle connected to two thermal baths, each with a temperature that is oscillating in time. Analytical expressions are derived for the energy fluxes in/out of the system and the baths, with excellent agreement observed between the analytical expressions and the results from nonequilibrium molecular dynamics simulations. We find that the shape of the heat transport hysteresis curves can be significantly altered by shifting the frequency between fast and slow oscillation regimes. We also observe the emergence of features in the hysteresis curves such as pinched loops and complex multi-loop patterns due to the frequency shifting. The presented results have implications in the design of thermal neuromorphic devices such as thermal memristors and thermal memcapacitors.

Transport and energetics of bacterial rectification

Randomly moving active particles can be herded into directed motion by asymmetric geometric structures. Although such a rectification process has been extensively studied due to its fundamental, biological, and technological relevance, a comprehensive understanding of active matter rectification based on single particle dynamics remains elusive. Here, by combining experiments, simulations, and theory, we study the directed transport and energetics of swimming bacteria navigating through funnel-shaped obstacles-a paradigmatic model of rectification of living active matter. We develop a microscopic parameter-free model for bacterial rectification, which quantitatively explains experimental and numerical observations and predicts the optimal geometry for the maximum rectification efficiency. Furthermore, we quantify the degree of time irreversibility and measure the extractable work associated with bacterial rectification. Our study provides quantitative solutions to long-standing questions on bacterial rectification and establishes a generic relationship between time irreversibility, particle fluxes, and extractable work, shedding light on the energetics of nonequilibrium rectification processes in living systems.

Fundamental Limits of an Irreversible Heat Engine

We investigated the optimal performance of an irreversible Stirling-like heat engine described by both overdamped and underdamped models within the framework of stochastic thermodynamics. By establishing a link between energy dissipation and Wasserstein distance, we derived the upper bound of maximal power that can be delivered over a complete engine cycle for both models. Additionally, we analytically developed an optimal control strategy to achieve this upper bound of maximal power and determined the efficiency at maximal power in the overdamped scenario.

Stochastic Model for a Piezoelectric Energy Harvester Driven by Broadband Vibrations

We present an experimental and numerical study of a piezoelectric energy harvester driven by broadband vibrations. This device can extract power from random fluctuations and can be described by a stochastic model, based on an underdamped Langevin equation with white noise, which mimics the dynamics of the piezoelectric material. A crucial point in the modelisation is represented by the appropriate description of the coupled load circuit that is necessary to harvest electrical energy. We consider a linear load (resistance) and a nonlinear load (diode bridge rectifier connected to the parallel of a capacitance and a load resistance), and focus on the characteristic curve of the extracted power as a function of the load resistance, in order to estimate the optimal values of the parameters that maximise the collected energy. In both cases, we find good agreement between the numerical simulations of the theoretical model and the results obtained in experiments. In particular, we observe a non-monotonic behaviour of the characteristic curve which signals the presence of an optimal value for the load resistance at which the extracted power is maximised. We also address a more theoretical issue, related to the inference of the non-equilibrium features of the system from data: we show that the analysis of high-order correlation functions of the relevant variables, when in the presence of nonlinearities, can represent a simple and effective tool to check the irreversible dynamics.

Principles of Molecular Evolution: Concepts from Non-equilibrium Thermodynamics for the Multilevel Theory of Learning

We present a non-equilibrium thermodynamics approach to the multilevel theory of learning for the study of molecular evolution. This approach allows us to study the explicit time dependence of molecular evolutionary processes and their impact on entropy production. Interpreting the mathematical expressions, we can show that two main contributions affect entropy production of molecular evolution processes which can be identified as mutation and gene transfer effects. Accordingly, our results show that the optimal adaptation of organisms to external conditions in the context of evolutionary processes is driven by principles of minimum entropy production. Such results can also be interpreted as the basis of some previous postulates of the theory of learning. Although our macroscopic approach requires certain simplifications, it allows us to interpret molecular evolutionary processes using thermodynamic descriptions with reference to well-known biological processes.

Dissipation Bounds Precision of Current Response to Kinetic Perturbations

The precision of currents in Markov networks is bounded by dissipation via the so-called thermodynamic uncertainty relation (TUR). In our Letter, we demonstrate a similar inequality that bounds the precision of the static current response to perturbations of kinetic barriers. Perturbations of such type, which affect only the system kinetics but not the thermodynamic forces, are highly important in biochemistry and nanoelectronics. We prove that our inequality cannot be derived from the standard TUR. Instead, it implies the standard TUR and provides an even tighter bound for dissipation. We also provide a procedure for obtaining the optimal response precision for a given model.