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We analyze the general relation between canonical and grand canonical ensembles in the thermodynamic limit. We begin our discussion by deriving, with an alternative approach, some standard results first obtained by Kac and coworkers in the late 1970s. Then, motivated by the Bose-Einstein condensation (BEC) of trapped gases with a fixed number of atoms, which is well described by the canonical ensemble and by the recent groundbreaking experimental realization of BEC with photons in a dye-filled optical microcavity under genuine grand canonical conditions, we apply our formalism to a system of non-interacting Bose particles confined in a two-dimensional harmonic trap. We discuss in detail the mathematical origin of the inequivalence of ensembles observed in the condensed phase, giving place to the so-called grand canonical catastrophe of density fluctuations. We also provide explicit analytical expressions for the internal energy and specific heat and compare them with available experimental data. For these quantities, we show the equivalence of ensembles in the thermodynamic limit.
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Absolute negative mobility (ANM) refers to the situation where the average velocity of a driven tracer is opposite to the direction of the driving force. This effect was evidenced in different models of nonequilibrium transport in complex environments, whose description remains effective. Here, we provide a microscopic theory for this phenomenon. We show that it emerges in the model of an active tracer particle submitted to an external force and which evolves on a discrete lattice populated with mobile passive crowders. Resorting to a decoupling approximation, we compute analytically the velocity of the tracer particle as a function of the different parameters of the system and confront our results to numerical simulations. We determine the range of parameters where ANM can be observed, characterize the response of the environment to the displacement of the tracer, and clarify the mechanism underlying ANM and its relationship with negative differential mobility (another hallmark of driven systems far from the linear response).
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We calculate the diffusion coefficient of an active tracer in a schematic crowded environment, represented as a lattice gas of passive particles with hardcore interactions. Starting from the master equation of the problem, we put forward a closure approximation that goes beyond trivial mean field and provides the diffusion coefficient for an arbitrary density of crowders in the system. We show that our approximation is accurate for a very wide range of parameters, and that it correctly captures numerous nonequilibrium effects, which are the signature of the activity in the system. In addition to the determination of the diffusion coefficient of the tracer, our approach allows us to characterize the perturbation of the environment induced by the displacement of the active tracer. Finally, we consider the asymptotic regimes of low and high densities, in which the expression of the diffusion coefficient of the tracer becomes explicit, and which we argue to be exact.
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Spontaneous brain activity is characterized by bursts and avalanche-like dynamics, with scale-free features typical of critical behaviour. The stochastic version of the celebrated Wilson-Cowan model has been widely studied as a system of spiking neurons reproducing non-trivial features of the neural activity, from avalanche dynamics to oscillatory behaviours. However, to what extent such phenomena are related to the presence of a genuine critical point remains elusive. Here we address this central issue, providing analytical results in the linear approximation and extensive numerical analysis. In particular, we present results supporting the existence of a bona fide critical point, where a second-order-like phase transition occurs, characterized by scale-free avalanche dynamics, scaling with the system size and a diverging relaxation time-scale. Moreover, our study shows that the observed critical behaviour falls within the universality class of the mean-field branching process, where the exponents of the avalanche size and duration distributions are, respectively, 3/2 and 2. We also provide an accurate analysis of the system behaviour as a function of the total number of neurons, focusing on the time correlation functions of the firing rate in a wide range of the parameter space.
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Encéfalo/fisiologia , Modelos Neurológicos , Potenciais de Ação/fisiologia , Animais , Biologia Computacional , Simulação por Computador , Fenômenos Eletrofisiológicos , Humanos , Modelos Lineares , Rede Nervosa/fisiologia , Neurônios/fisiologia , Análise Espaço-Temporal , Processos EstocásticosRESUMO
We study the power extracted by an electromagnetic energy harvester driven by broadband vibrations. We describe the system with a linear model, featuring an underdamped stochastic differential equation for an effective mass in a harmonic potential, coupled electromechanically with the current in the circuit. We compare the characteristic curve (power vs. load resistance) obtained in experiments for several values of the vibration amplitude with the analytical results computed from the model. Then, we focus on a more refined analysis, taking into account the temporal correlations of the current signal and the fluctuations of the extracted power over finite times. We find a very good agreement between the analytical predictions and the experimental data, showing that the linear model with effective parameters can describe the real system, even at the fine level of fluctuations. Our results could be useful in the framework of stochastic thermodynamics applied to energy harvesting systems.
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We experimentally study a piezoelectric energy harvester driven by broadband random vibrations. We show that a linear model, consisting of an underdamped Langevin equation for the dynamics of the tip mass, electromechanically coupled with a capacitor and a load resistor, can accurately describe the experimental data. In particular, the theoretical model allows us to define fluctuating currents and to study the stochastic thermodynamics of the system, with focus on the distribution of the extracted work over different time intervals. Our analytical and numerical analysis of the linear model is succesfully compared to the experiments.
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We study the dynamics of a self-propelled particle advected by a steady laminar flow. The persistent motion of the self-propelled particle is described by an active Ornstein-Uhlenbeck process. We focus on the diffusivity properties of the particle as a function of persistence time and free-diffusion coefficient, revealing non-monotonic behaviors, with the occurrence of a minimum and a steep growth in the regime of large persistence time. In the latter limit, we obtain an analytical prediction for the scaling of the diffusion coefficient with the parameters of the active force. Our study sheds light on the effect of a flow-field on the diffusion of active particles, such as living microorganisms and motile phytoplankton in fluids.
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Modelos Teóricos , Simulação por Computador , Difusão , Movimento (Física)RESUMO
In this paper we review some general properties of probability distributions which exhibit a singular behavior. After introducing the matter with several examples based on various models of statistical mechanics, we discuss, with the help of such paradigms, the underlying mathematical mechanism producing the singularity and other topics such as the condensation of fluctuations, the relationships with ordinary phase-transitions, the giant response associated to anomalous fluctuations, and the interplay with fluctuation relations.
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We study the diffusion of a tracer particle driven out of equilibrium by an external force and traveling in a dense environment of arbitrary density. The system evolves on a discrete lattice and its stochastic dynamics is described by a master equation. Relying on a decoupling approximation that goes beyond the naive mean-field treatment of the problem, we calculate the fluctuations of the position of the tracer around its mean value on a lattice of arbitrary dimension, and with different boundary conditions. We reveal intrinsically nonequilibrium effects, such as enhanced diffusivity of the tracer induced by both the crowding interactions and the external driving. We finally consider the high-density and low-density limits of the model and show that our approximation scheme becomes exact in these limits.
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A challenging frontier in modern statistical physics is concerned with systems with a small number of degrees of freedom, far from the thermodynamic limit.[...].
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The rectification of unbiased fluctuations, also known as the ratchet effect, is normally obtained under statistical nonequilibrium conditions. Here we propose a new ratchet mechanism where a thermal bath solicits the random rotation of an asymmetric wheel, which is also subject to Coulomb friction due to solid-on-solid contacts. Numerical simulations and analytical calculations demonstrate a net drift induced by friction. If the thermal bath is replaced by a granular gas, the well-known granular ratchet effect also intervenes, becoming dominant at high collision rates. For our chosen wheel shape the granular effect acts in the opposite direction with respect to the friction-induced torque, resulting in the inversion of the ratchet direction as the collision rate increases. We have realized a new granular ratchet experiment where both these ratchet effects are observed, as well as the predicted inversion at their crossover. Our discovery paves the way to the realization of micro and submicrometer Brownian motors in an equilibrium fluid, based purely upon nanofriction.
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The relation between spontaneous and stimulated brain activity is a fundamental question in neuroscience which has received wide attention in experimental studies. Recently, it has been suggested that the evoked response to external stimuli can be predicted from temporal correlations of spontaneous activity. Previous theoretical results, confirmed by the comparison with magnetoencephalography data for human brains, were obtained for the Wilson-Cowan model in the condition of balance of excitation and inhibition, a signature of a healthy brain. Here we extend previous studies to imbalanced conditions by examining a region of parameter space around the balanced fixed point. Analytical results are compared to numerical simulations of Wilson-Cowan networks. We evidence that in imbalanced conditions the functional form of the time correlation and response functions can show several behaviors, exhibiting also an oscillating regime caused by the emergence of complex eigenvalues. The analytical predictions are fully in agreement with numerical simulations, validating the role of cross-correlations in the response function. Furthermore, we identify the leading role of inhibitory neurons in controlling the overall activity of the system, tuning the level of excitability and imbalance.
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Encéfalo , Neurônios , Humanos , Neurônios/fisiologia , Encéfalo/fisiologiaRESUMO
Fluctuating entropy production is studied for a set of linearly coupled complex fields. The general result is applied to non-equilibrium fluctuating hydrodynamic equations for coarse-grained fields (density, temperature, and velocity), in the framework of model granular fluids. We find that the average entropy production, obtained from the microscopic stochastic description, can be expressed in terms of macroscopic quantities, in analogy with linear non-equilibrium thermodynamics. We consider the specific cases of driven granular fluids with two different kinds of thermostat and the homogeneous cooling regime. In all cases, the average entropy production turns out to be the product of a thermodynamic force and a current: the former depends on the specific energy injection mechanism, the latter takes always the form of a static correlation between fluctuations of density and temperature time-derivative. Both vanish in the elastic limit. The behavior of the entropy production is studied at different length scales and the qualitative differences arising for the different granular models are discussed.
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Velocity and density structure factors are measured over a hydrodynamic range of scales in a horizontal quasi-2D fluidized granular experiment, with packing fractions φ ∈ [10%, 40%]. The fluidization is realized by vertically vibrating a rough plate, on top of which particles perform a Brownian-like horizontal motion in addition to inelastic collisions. On one hand, the density structure factor is equal to that of elastic hard spheres, except in the limit of large length-scales, as it occurs in the presence of an effective interaction. On the other hand, the velocity field shows a more complex structure which is a genuine expression of a non-equilibrium steady state and which can be compared to a recent fluctuating hydrodynamic theory with non-equilibrium noise. The temporal decay of velocity modes autocorrelations is compatible with linear hydrodynamic equations with rates dictated by viscous momentum diffusion, corrected by a typical interaction time with the thermostat. Equal-time velocity structure factors display a peculiar shape with a plateau at large length-scales and another one at small scales, marking two different temperatures: the "bath" temperature T(b), depending on shaking parameters, and the "granular" temperature T(g) < T(b), which is affected by collisions. The two ranges of scales are separated by a correlation length which grows with φ, after proper rescaling with the mean free path.
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Many systems in nature exhibit avalanche dynamics with scale-free features. A general scaling theory has been proposed for critical avalanche profiles in crackling noise, predicting the collapse onto a universal avalanche shape, as well as the scaling behavior of the activity power spectrum as Brown noise. Recently, much attention has been given to the profile of neuronal avalanches, measured in neuronal systems in vitro and in vivo. Although a universal profile was evidenced, confirming the validity of the general scaling theory, the parallel study of the power spectrum scaling under the same conditions was not performed. The puzzling observation is that in the majority of healthy neuronal systems the power spectrum exhibits a behavior close to 1/f, rather than Brown, noise. Here we perform a numerical study of the scaling behavior of the avalanche shape and the power spectrum for a model of integrate and fire neurons with a short-term plasticity parameter able to tune the system to criticality. We confirm that, at criticality, the average avalanche size and the avalanche profile fulfill the general avalanche scaling theory. However, the power spectrum consistently exhibits Brown noise behavior, for both fully excitatory networks and systems with 30% inhibitory networks. Conversely, a behavior closer to 1/f noise is observed in systems slightly off criticality. Results suggest that the power spectrum is a good indicator to determine how close neuronal activity is to criticality.
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A unified derivation of the off-equilibrium fluctuation dissipation relations (FDRs) is given for Ising and continuous spins to arbitrary order, within the framework of Markovian stochastic dynamics. Knowledge of the FDRs allows one to develop zero field algorithms for the efficient numerical computation of the response functions. Two applications are presented. In the first one, the problem of probing for the existence of a growing cooperative length scale is considered in those cases, like in glassy systems, where the linear FDR is of no use. The effectiveness of an appropriate second order FDR is illustrated in the test case of the Edwards-Anderson spin glass in one and two dimensions. In the second application, the important problem of the definition of an off-equilibrium effective temperature through the nonlinear FDR is considered. It is shown that, in the case of coarsening systems, the effective temperature derived from the second order FDR is consistent with the one obtained from the linear FDR.
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Granular media take on great importance in industry and geophysics, posing a severe challenge to materials science. Their response properties elude known soft rheological models, even when the yield-stress discontinuity is blurred by vibro-fluidization. Here we propose a broad rheological scenario where average stress sums up a frictional contribution, generalizing conventional µ(I)-rheology, and a kinetic collisional term dominating at fast fluidization. Our conjecture fairly describes a wide series of experiments in a vibrofluidized vane setup, whose phenomenology includes velocity weakening, shear thinning, a discontinuous thinning transition, and gaseous shear thickening. The employed setup gives access to dynamic fluctuations, which exhibit a broad range of timescales. In the slow dense regime the frequency of cage-opening increases with stress and enhances, with respect to µ(I)-rheology, the decrease of viscosity. Diffusivity is exponential in the shear stress in both thinning and thickening regimes, with a huge growth near the transition.
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The condition of thermal equilibrium simplifies the theoretical treatment of fluctuations as found in the celebrated Einstein's relation between mobility and diffusivity for Brownian motion. Several recent theories relax the hypothesis of thermal equilibrium resulting in at least two main scenarios. With well separated timescales, as in aging glassy systems, equilibrium Fluctuation-Dissipation Theorem applies at each scale with its own "effective" temperature. With mixed timescales, as for example in active or granular fluids or in turbulence, temperature is no more well-defined, the dynamical nature of fluctuations fully emerges and a Generalized Fluctuation-Dissipation Theorem (GFDT) applies. Here, we study experimentally the mixed timescale regime by studying fluctuations and linear response in the Brownian motion of a rotating intruder immersed in a vibro-fluidized granular medium. Increasing the packing fraction, the system is moved from a dilute single-timescale regime toward a denser multiple-timescale stage. Einstein's relation holds in the former and is violated in the latter. The violation cannot be explained in terms of effective temperatures, while the GFDT is able to impute it to the emergence of a strong coupling between the intruder and the surrounding fluid. Direct experimental measurements confirm the development of spatial correlations in the system when the density is increased.
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Difusão , Movimento (Física) , Modelos Químicos , Temperatura , TermodinâmicaRESUMO
We report the study of an experimental granular Brownian motor, inspired by the one published in Eshuis et al. [Phys. Rev. Lett. 104, 248001 (2010)], but different in some ingredients. As in that previous work, the motor is constituted by a rotating blade, the surfaces of which break the rotation-inversion symmetry through alternated patches of different inelasticity, immersed in a gas of granular particles. The main difference of our experimental setup is in the orientation of the main axis, which is parallel to the (vertical) direction of shaking of the granular fluid, guaranteeing an isotropic distribution for the velocities of colliding grains, characterized by a variance v(0)(2). We also keep the granular system diluted, in order to compare with Boltzmann-equation-based kinetic theory. In agreement with theory, we observe the crucial role of Coulomb friction which induces two main regimes: (i) rare collisions, with an average angular velocity <ω>~v(0)(3), and (ii) frequent collisions (FC), with <ω>~v(0). We also study the fluctuations of the angle spanned in a large-time interval Δθ, which in the FC regime is proportional to the work done upon the motor. We observe that the fluctuation relation is satisfied with a slope which weakly depends on the relative collision frequency.
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Coloides/química , Difusão , Transferência de Energia , Modelos Químicos , Modelos Moleculares , Termodinâmica , Simulação por Computador , Fricção , Cinética , Movimento (Física)RESUMO
We discuss the relation between the fluctuation-dissipation relation derived by Chatelain and Ricci-Tersenghi [C. Chatelain, J. Phys. A 36, 10739 (2003); F. Ricci-Tersenghi, Phys. Rev. E 68, 065104(R) (2003)] and that by Lippiello-Corberi-Zannetti [E. Lippiello, F. Corberi, and M. Zannetti, Phys. Rev. E 71, 036104 (2005)]. In order to do that, we rederive the fluctuation-dissipation relation for systems of discrete variables evolving in discrete time via a stochastic nonequilibrium Markov process. The calculation is carried out in a general formalism comprising the Chatelain, Ricci-Tersenghi, result and that by Lippiello-Corberi-Zannetti as special cases. The applicability, generality, and experimental feasibility of the two approaches are thoroughly discussed. Extending the analytical calculation to the variance of the response function, we show the advantage of field-free numerical methods with respect to the standard method, where the perturbation is applied. We also show that the signal-to-noise ratio is better (by a factor square root of 2) in the algorithm of Lippiello-Corberi-Zannetti with respect to that of Chatelain-Ricci Tersenghi.