Fluctuation-response Relation Research Articles

Active fluctuations in the harmonic chain: phonons, entropons and velocity correlations

Non-equilibrium random fluctuations of a non-thermal nature are a defining feature of active matter. In this work, we study the collective excitations of active systems at high density, focusing on a one-dimensional chain of elastically coupled inertial particles, where activity is modeled by an Ornstein–Uhlenbeck process. The excitation spectrum reveals two types of fluctuations: thermally excited phonons, analogous to those in passive crystals, and entropons, which are associated with entropy production due to active forces. These fluctuations exhibit distinct properties: only entropons generate spatial velocity correlations and violate the standard fluctuation-response relation. We derive exact expressions for equal-time velocity and displacement correlations, as well as for the structure factor, identifying the contributions from both phonons and entropons. Finally, we explore the dynamical properties of these excitations through steady-state two-time correlations, such as the intermediate scattering function and mean-square displacement. Both phonon and entropon fluctuations are characterized by a long-wavelength overdamped regime and a short-wavelength underdamped regime. In the large persistence case, entropons decay more slowly than phonons, and activity generally suppresses the oscillations typical of the underdamped regime.

Fluctuation-response relation as a probe of long-range correlations in nonequilibrium quantum and classical fluids.

The absence of a simple fluctuation-dissipation theorem is a major obstacle for studying systems that are not in thermodynamic equilibrium. We show that for a fluid in a nonequilibrium steady state characterized by a constant temperature gradient the commutator correlation functions are still related to response functions; however, the relation is to the bilinear response of products of two observables, rather than to a single linear response function as is the case in equilibrium. This modified fluctuation-response relation holds for both quantum and classical systems. It is both motivated and informed by the long-range correlations that exist in such a steady state and allows for probing them via response experiments. This is of particular interest in quantum fluids, where the direct observation of fluctuations by light scattering would be difficult. In classical fluids it is known that the coupling of the temperature gradient to the diffusive shear velocity leads to correlations of various observables, in particular temperature fluctuations, that do not decay as a function of distance, but rather extend over the entire system. We investigate the nature of these correlations in a fermionic quantum fluid and show that the crucial coupling between the temperature gradient and velocity fluctuations is the same as in the classical case. Accordingly, the nature of the long-ranged correlations in the hydrodynamic regime also is the same. However, as one enters the collisionless regime in the low-temperature limit the nature of the velocity fluctuations changes: they become ballistic rather than diffusive. As a result, correlations of the temperature and other observables are still singular in the long-wavelength limit, but the singularity is weaker than in the hydrodynamic regime.

Fluctuation-response relations for integrate-and-fire models with an absolute refractory period.

We study the problem of relating the spontaneous fluctuations of a stochastic integrate-and-fire (IF) model to the response of the instantaneous firing rate to time-dependent stimulation if the IF model is endowed with a non-vanishing refractory period and a finite (stereotypical) spike shape. This seemingly harmless addition to the model is shown to complicate the analysis put forward by Lindner Phys. Rev. Lett. (2022), i.e., the incorporation of the reset into the model equation, the Rice-like averaging of the stochastic differential equation, and the application of the Furutsu-Novikov theorem. We derive a still exact (although more complicated) fluctuation-response relation (FRR) for an IF model with refractory state and a white Gaussian background noise. We also briefly discuss an approximation for the case of a colored Gaussian noise and conclude with a summary and outlook on open problems.

Nonintegral form of the reciprocal relation associated with violation of the fluctuation response relation.

We extend Onsager's reciprocal relation to systems in a nonequilibrium steady state. While Onsager's reciprocal relation concerns the kinetic (Onsager) coefficient, the extended reciprocal relation concerns violation of the fluctuation response relation (FRR) for mechanical and thermal perturbations. This extended relation holds at each frequency when the extent of the FRR violation is expressed in a frequency domain. This nonintegral form distinguishes the extended relation from previous relations expressed by integration over a frequency. To obtain this relation, we consider one-particle one-dimensional systems described by an overdamped Langevin equationwith a force driving the system away from equilibrium. We assume a special property of the potential in the system. From this Langevin equation, we obtain the Fokker-Planck (FP) equationdescribing the time evolution of the distribution function of the particle. Using the FP equation, we calculate the responses of the particle velocity and heat current by applying time-dependent perturbations of the driving force and temperature. We express the extent of the FRR violation in terms of these responses with time correlation functions and expand them in powers of the FP operator. This reciprocal relation is valid far from equilibrium. One can also confirm this reciprocal relation through experiments with systems such as colloidal suspensions because the FRR violation can be experimentally observed.

Langevin analogy between particle trajectories and polymer configurations.

A diffusive trajectory drawn by the generalized Langevin equation(GLE) for a colloidal particle evokes a random fractal of a static polymer configuration. This article proposes a static GLE-like description that enables the generation of a single configuration of a polymer chain with the noise formulated to satisfy the static fluctuation-response relation (FRR) along a one-dimensional chain structure but not along a temporal coordinate. A remarkable point is qualitative differences and similarities in the FRR formulation between the static and the dynamic GLEs. Guided by the static FRR, we further make analogous arguments in light of stochastic energetics and the steady-state fluctuation theorem.

Dynamical fluctuations of a tracer coupled to active and passive particles

We study the induced dynamics of an inertial tracer particle elastically coupled to passive or active Brownian particles. We integrate out the environment degrees of freedom to obtain the exact effective equation of the tracer—a generalized Langevin equation in both cases. In particular, we find the exact form of the dissipation kernel and effective noise experienced by the tracer and compare it with the phenomenological modeling of active baths used in previous studies. We show that the second fluctuation-dissipation relation (FDR) does not hold at early times for both cases. However, at finite times, the tracer dynamics violate (obeys) the FDR for the active (passive) environment. We calculate the linear response formulas in this regime for both cases and show that the passive medium satisfies an equilibrium fluctuation response relation, while the active medium does not—we quantify the extent of this violation explicitly. We show that though the active medium generally renders a nonequilibrium description of the tracer, an effective equilibrium picture emerges asymptotically in the small activity limit of the medium. We also calculate the mean squared velocity and mean squared displacement of the tracer and report how they vary with time.

Fluctuation-response relation of time-symmetric quantities around general nonequilibrium stationary state

A connection between the response and fluctuation in general nonequilibrium stationary states is investigated. We focus on time-symmetric quantities and find that the fluctuation of a kind of empirical measure can be expressed with the response of the empirical measure, current, and the time-symmetric current. This relation is proven by using the fictitious stalling decomposition: we decompose a single observed transition (edge in the state space) between two microscopic states into two transitions such that one of the transitions stalls in this stationary state. Through this trick, relations for stalling stationary states apply to general nonequilibrium stationary states, which leads to the desired relation.

Non-equilibrium dynamics of bacterial colonies—growth, active fluctuations, segregation, adhesion, and invasion

Colonies of bacteria endowed with a pili-based self-propulsion machinery are ideal models for investigating the structure and dynamics of active many-particle systems. We study Neisseria gonorrhoeae colonies with a molecular-dynamics-based approach. A generic, adaptable simulation method for particle systems with fluctuating bond-like interactions is devised. The simulations are employed to investigate growth of bacterial colonies and the dependence of the colony structure on cell-cell interactions. In colonies, pilus retraction enhances local ordering. For colonies consisting of different types of cells, the simulations show a segregation depending on the pili-mediated interactions among different cells. These results agree with experimental observations. Next, we quantify the power-spectral density of colony-shape fluctuations in silico. Simulations predict a strong violation of the equilibrium fluctuation-response relation. Furthermore, we show that active force generation enables colonies to spread on surfaces and to invade narrow channels. The methodology can serve as a foundation for future studies of active many-particle systems at boundaries with complex shape.

Time-Symmetric Current and Its Fluctuation Response Relation around Nonequilibrium Stalling Stationary State.

We propose a time-symmetric counterpart of the current in stochastic thermodynamics named the time-symmetric current. This quantity is defined with empirical measures and thus is symmetric under time reversal, while its ensemble average reproduces the amount of the average current. We prove that this time-symmetric current satisfies the fluctuation-response relation in the conventional form but with sign inversion. Remarkably, this fluctuation-response relation holds not only around equilibrium states but also around nonequilibrium stationary states if observed currents stall. The obtained relation also serves as an experimental tool for probing the value of a bare transition rate by measuring only time-integrated empirical measures.

The inverse variance–flatness relation in stochastic gradient descent is critical for finding flat minima

Despite tremendous success of the stochastic gradient descent (SGD) algorithm in deep learning, little is known about how SGD finds generalizable solutions at flat minima of the loss function in high-dimensional weight space. Here, we investigate the connection between SGD learning dynamics and the loss function landscape. A principal component analysis (PCA) shows that SGD dynamics follow a low-dimensional drift-diffusion motion in the weight space. Around a solution found by SGD, the loss function landscape can be characterized by its flatness in each PCA direction. Remarkably, our study reveals a robust inverse relation between the weight variance and the landscape flatness in all PCA directions, which is the opposite to the fluctuation-response relation (aka Einstein relation) in equilibrium statistical physics. To understand the inverse variance-flatness relation, we develop a phenomenological theory of SGD based on statistical properties of the ensemble of minibatch loss functions. We find that both the anisotropic SGD noise strength (temperature) and its correlation time depend inversely on the landscape flatness in each PCA direction. Our results suggest that SGD serves as a landscape-dependent annealing algorithm. The effective temperature decreases with the landscape flatness so the system seeks out (prefers) flat minima over sharp ones. Based on these insights, an algorithm with landscape-dependent constraints is developed to mitigate catastrophic forgetting efficiently when learning multiple tasks sequentially. In general, our work provides a theoretical framework to understand learning dynamics, which may eventually lead to better algorithms for different learning tasks.

Nonequilibrium grand-canonical ensemble built from a physical particle reservoir

We introduce a nonequilibrium grand-canonical ensemble defined by considering the stationary state of a driven system of particles put in contact with a particle reservoir. When an additivity assumption holds for the large deviation function of density, a chemical potential of the reservoir can be defined. The grand-canonical distribution then takes a form similar to the equilibrium one. At variance with equilibrium, though, the probability weight is "renormalized" by a contribution coming from the contact, with respect to the canonical probability weight of the isolated system. A formal grand-canonical potential can be introduced in terms of a scaled cumulant generating function, defined as the Legendre-Fenchel transform of the large deviation function of density. The role of the formal Legendre parameter can be played, physically, by the chemical potential of the reservoir when the latter can be defined, or by a potential energy difference applied between the system and the reservoir. Static fluctuation-response relations naturally follow from the large deviation structure. Some of the results are illustrated on two different explicit examples, a gas of noninteracting active particles and a lattice model of interacting particles.

Transport fluctuation relations in interacting quantum pumps

The understanding of out-of-equilibrium fluctuation relations in small open quantum systems has been a focal point of research in recent years. In particular, for systems with adiabatic time-dependent driving, it was shown that the fluctuation relations known from stationary systems do no longer apply due the geometric nature of the pumping current response. However, the precise physical interpretation of the corrected pumping fluctuation relations as well as the role of many-body interactions remained unexplored. Here, we study quantum systems with many-body interactions subject to slow time-dependent driving, and show that fluctuation relations of the charge current can in general not be formulated without taking into account the total energy current put into the system through the pumping process. Moreover, we show that this correction due to the input energy is nonzero only when Coulomb-interactions are present. Thus, fluctuation response relations offer an until now unrevealed opportunity to probe many-body correlations in quantum systems. We demonstrate our general findings at the concrete example of a single-level quantum dot model, and propose a scheme to measure the interaction-induced discrepancies from the stationary case.

Sub-Gaussian and subexponential fluctuation-response inequalities.

Sub-Gaussian and subexponential distributions are introduced and applied to study the fluctuation-response relation out of equilibrium. A bound on the difference in expected values of an arbitrary sub-Gaussian or subexponential physical quantity is established in terms of its sub-Gaussian or subexponential norm. Based on that, we find that the entropy difference between two states is bounded by the energy fluctuation in these states. Moreover, we obtain generalized versions of the thermodynamic uncertainty relation in different regimes. Operational issues concerning the application of our results in an experimental setting are also addressed, and nonasymptotic bounds on the errors incurred by using the sample mean instead of the expected value in our fluctuation-response inequalities are derived.

Correlations and responses for a system of n coupled linear oscillators with asymmetric interactions.

We focus on the asymmetry of the interaction in the optimal velocity (OV) model, which is a model of self-driven particles, and analytically investigate the effects of the asymmetry on the fluctuation-response relation, which is one of the remarkable relationships in statistical physics. By linearizing a modified OV model, i.e., the backward-looking optimal velocity model, which can easily control the magnitude of asymmetry in the interaction, we derive n coupled linear oscillators with asymmetric interactions. We analytically solve the equations of the n coupled linear oscillators and calculate the response and correlation functions. We find that the fluctuation response relation does not hold in the n coupled linear oscillators with asymmetric interactions. Moreover, as the magnitude of the asymmetry increases, the difference between the response and correlation functions increases .

Additivity and density fluctuations in Vicsek-like models of self-propelled particles.

We study coarse-grained density fluctuations in the disordered phase of the paradigmatic Vicsek-like models of self-propelled particles with alignment interactions and random self-propulsion velocities. By numerically integrating a fluctuation-response relation-the direct consequence of an additivity property-we compute logarithm of the large-deviation probabilities of the coarse-grained subsystem density, while the system is in the disordered fluid phase with vanishing macroscopic velocity. The large-deviation probabilities, computed within additivity, agree remarkably well with that obtained from direct microscopic simulations of the models. Our results provide evidence of the existence of an equilibriumlike chemical potential, which governs the coarse-grained density fluctuations in the Vicsek-like models. Moreover, comparison of the particle-number fluctuations among several self-propelled particle systems suggests a common mechanism through which the number fluctuations arise in such systems.

Measuring Dissipation of Molecular Motor Kinesin

Molecular motors are nonequilibrium open systems that convert chemical energy to mechanical work. The nonequilibrium energetics of single molecule kinesin were investigated by measuring the motion of an attached probe particle and its response to external forces with optical tweezers. The sum of the heat dissipation estimated from the violation of the fluctuation-response relation and the output power was inconsistent with the input free energy rate, indicating that large internal dissipation exists. Here we introduce the theoretical basis of the dissipation measurement and our recent experimental results, discussing the physiological implications of the hidden dissipation in the kinesin motor.

Nonequilibrium Energetics of Molecular Motor Kinesin.

Nonequilibrium energetics of single molecule translational motor kinesin was investigated by measuring heat dissipation from the violation of the fluctuation-response relation of a probe attached to the motor using optical tweezers. The sum of the dissipation and work did not amount to the input free energy change, indicating large hidden dissipation exists. Possible sources of the hidden dissipation were explored by analyzing the Langevin dynamics of the probe, which incorporates the two-state Markov stepper as a kinesin model. We conclude that internal dissipation is dominant.

Reciprocal relations associated with linear responses to mechanical and thermal perturbations in nonequilibrium Langevin systems

We study universal relationships in linear responses to instances when mechanical and thermal perturbations are applied to a nonequilibrium steady state. For perturbations in nonequilibrium Langevin systems, we consider a time-dependent external force exerted on a particle along with a time-dependent temperature for the heat bath. In this system, by calculating heat currents and particle velocity, we obtain the kinetic coefficients (Onsager's coefficients) and the time correlation functions using the Fokker--Planck equations. With these quantities, we derive the reciprocal relation associated with the violation of the fluctuation response relation. We obtain the relation by integrating the extent of the violation with respect to the frequency. We also find the reciprocity in the first moment of the frequency integration. These frequency sum rules hold for both overdamped and underdamped regimes for any type of potential and any strength of the external force.

Two-tag correlations and nonequilibrium fluctuation–response relation in ageing single-file diffusion

Spatiotemporally correlated motions of interacting Brownian particles, confined in a narrow channel of infinite length, are studied in terms of statistical quantities involving two particles. A theoretical framework that allows analytical calculation of two-tag correlations is presented on the basis of the Dean–Kawasaki equation describing density fluctuations in colloidal systems. In the equilibrium case, the time-dependent Einstein relation holds between the two-tag displacement correlation and the response function corresponding to it, which is a manifestation of the fluctuation–dissipation theorem for the correlation of density fluctuations. While the standard procedure of closure approximation for nonlinear density fluctuations is known to be obstructed by inconsistency with the fluctuation–dissipation theorem, this difficulty is naturally avoided by switching from the standard Fourier representation of the density field to the label-based Fourier representation of the vacancy field. In the case of ageing dynamics started from equidistant lattice configuration, the time-dependent Einstein relation is violated, as the two-tag correlation depends on the waiting time for equilibration while the response function is not sensitive to it. Within linear approximation, however, there is a simple relation between the density (or vacancy) fluctuations and the corresponding response function, which is valid even if the system is out of equilibrium. This non-equilibrium fluctuation–response relation can be extended to the case of nonlinear fluctuations by means of closure approximation for the vacancy field.

Thermal response of a Fermi\u2013Pasta\u2013Ulam chain with Andersen thermostats

The linear response to temperature variations is well characterised for equilibrium systems but a similar theory is not available, for example, for inertial heat conducting systems, whose paradigm is the Fermi–Pasta–Ulam (FPU) model driven by two different boundary temperatures. For models of inertial systems out of equilibrium, including relaxing systems, we show that Andersen thermostats are a natural tool for studying the thermal response. We derive a fluctuation-response relation that allows to predict thermal expansion coefficients or the heat capacitance in nonequilibrium regimes. Simulations of the FPU chain of oscillators suggest that estimates of susceptibilities obtained with our relation are better than those obtained via a small perturbation.