arXiv subjects

Domingos S. P. Salazar

Publications and source records attributed to Domingos S. P. Salazar.

36 records · Page 2Linked to original sources

arXiv2023

Improving the Cramér-Rao bound with the detailed fluctuation theorem

In some non-equilibrium systems, the distribution of entropy production $p(Σ)$ satisfies the detailed fluctuation theorem (DFT), $p(Σ)/p(-Σ)=\exp(Σ)$. When the distribution $p(Σ)$ shows time-dependency, the celebrated Cramér-Rao (CR) bound asserts that the mean entropy production rate is upper bounded in terms of the variance of $Σ$ and the Fisher information with respect to time. In this letter, we employ the DFT to derive an upper bound for the mean entropy production rate that improves the CR bound. We show that this new bound serves as an accurate approximation for the entropy production rate in the heat exchange problem mediated by a weakly coupled bosonic mode. The bound is saturated for the same setup when mediated by a weakly coupled qubit.

cond-mat.stat-mech↗

arXiv2023

Thermodynamic variational relation

In systems far from equilibrium, the statistics of observables are connected to entropy production, leading to the Thermodynamic Uncertainty Relation (TUR). However, the derivation of TURs often involves constraining the parity of observables, such as considering asymmetric currents, making it unsuitable for the general case. We propose a Thermodynamic Variational Relation (TVR) between the statistics of general observables and entropy production, based on the variational representation of $f$-divergences. From this result, we derive a universal TUR and other relations for higher-order statistics of observables.

cond-mat.stat-mech↗

arXiv2023

Moment generating function bound from detailed fluctuation theorem

A famous consequence of the detailed fluctuation theorem (FT), $p(Σ)/p(-Σ)=\exp{(Σ)}$, is the integral FT $\langle \exp(-Σ)\rangle =1$ for a random variable $Σ$ and a distribution $p(Σ)$. When $Σ$ represents the entropy production in thermodynamics, the main outcome of the integral FT is the second law, $\langle Σ\rangle \geq 0$. However, a full description of the fluctuations of $Σ$ might require knowledge of the moment generating function (MGF), $G(α):=\langle \exp(αΣ) \rangle$. In the context of the detailed FT, we show the MGF is lower bounded in the form $G(α)\geq B(α,\langleΣ\rangle)$ for a given mean $\langleΣ\rangle$. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a weakly coupled bosonic mode and a qubit swap engine.

cond-mat.stat-mech↗

arXiv2022

Thermodynamic uncertainty relation from involutions

The Thermodynamic Uncertainty Relation (TUR) is a lower bound for the variance of a current as a function of the average entropy production and average current. Depending on the assumptions, one obtains different versions of the TUR. For instance, from the exchange fluctuation theorem, one obtains a corresponding exchange TUR. Alternatively, we show that TURs are a consequence of a very simple property: every process $s$ has only one inverse $s'=m(s)$, where $m$ is an involution, $m(m(s))=s$. This property allows the derivation of a general TUR without using any fluctuation theorem. As applications, we obtain the exchange TUR, the asymmetric TUR, a fluctuation-response inequality and lower bounds for the entropy production in terms of other nonequilibrium metrics.

cond-mat.stat-mech↗

arXiv2022

Thermodynamic Skewness Relation From Detailed Fluctuation Theorem

The detailed fluctuation theorem (DFT) is a statement about the asymmetry in the statistics of the entropy production. Consequences of the DFT are the second law of thermodynamics and the thermodynamics uncertainty relation (TUR), which translate into lower bounds for the mean and variance of currents, respectively. However, far from equilibrium, mean and variance are not enough to characterize the underlying distribution of the entropy production. The fluctuations are not necessarily Gaussian (nor symmetric), which means its skewness could be nonzero. We prove that the DFT imposes a negative tight lower bound for the skewness of the entropy production as a function of the mean. As application, we check the bound in the heat exchange problem between two thermal reservoirs mediated by a qubit swap engine.

cond-mat.stat-mech↗

arXiv2022

Lower bound for entropy production rate in stochastic systems far from equilibrium

We show that the Schnakenberg's entropy production rate in a master equation is lower bounded by a function of the weight of the Markov graph, here defined as the sum of the absolute values of probability currents over the edges. The result is valid for time-dependent nonequilibrium entropy production rates. Moreover, in a general framework, we prove a theorem showing that the Kullback-Leibler divergence between distributions $P(s)$ and $P'(s):=P(m(s))$, where $m$ is an involution, $m(m(s))=s$, is lower bounded by a function of the total variation of $P$ and $P'$, for any $m$. The bound is tight and it improves on Pinsker's inequality for this setup. This result illustrates a connection between nonequilibrium thermodynamics and graph theory with interesting applications.

cond-mat.stat-mech↗

arXiv2022

Upper bound for quantum entropy production from entropy flux

Entropy production is a key quantity characterizing nonequilibrium systems. However, it can often be difficult to compute in practice, as it requires detailed information about the system and the dynamics it undergoes. This becomes even more difficult in the quantum domain, and if one is interested in generic nonequilibrium reservoirs, for which the standard thermal recipes no longer apply. In this paper, we derive an upper bound for the entropy production in terms of the entropy flux for a class of systems for which the flux is given in terms of a system's observable. Since currents are often easily accessible in this class of systems, this bound should prove useful for estimating the entropy production in a broad variety of processes. We illustrate the applicability of the bound by considering a three-level maser engine and a system interacting with a squeezed bath.

quant-ph↗

arXiv2021

Detailed fluctuation theorem bounds apparent violations of the second law

The second law of thermodynamics is a statement about the statistics of the entropy production, $\langle Σ\rangle \geq 0$. For small systems, it is known that the entropy production is a random variable and negative values ($Σ< 0$) might be observed in some experiments. This situation is sometimes called apparent violation of the second law. In this sense, how often is the second law violated? For a given average $\langle Σ\rangle $, we show that the strong detailed fluctuation theorem implies a lower tight bound for the apparent violations of the second law. As applications, we verify that the bound is satisfied for the entropy produced in the heat exchange problem between two reservoirs mediated by a bosonic mode in the weak coupling approximation, a levitated nanoparticle and a classical particle in a box.

cond-mat.stat-mech↗

arXiv2021

Nonequilibrium thermodynamics of self-supervised learning

Self-supervised learning (SSL) of energy based models has an intuitive relation to equilibrium thermodynamics because the softmax layer, mapping energies to probabilities, is a Gibbs distribution. However, in what way SSL is a thermodynamic process? We show that some SSL paradigms behave as a thermodynamic composite system formed by representations and self-labels in contact with a nonequilibrium reservoir. Moreover, this system is subjected to usual thermodynamic cycles, such as adiabatic expansion and isochoric heating, resulting in a generalized Gibbs ensemble (GGE). In this picture, we show that learning is seen as a demon that operates in cycles using feedback measurements to extract negative work from the system. As applications, we examine some SSL algorithms using this idea.

cond-mat.stat-mech↗

arXiv2021

Information bound for entropy production from the detailed fluctuation theorem

Fluctuation theorems impose fundamental bounds in the statistics of the entropy production, with the second law of thermodynamics being the most famous. Using information theory, we quantify the information of entropy production and find an upper tight bound as a function of its mean from the strong detailed fluctuation theorem. The bound is given in terms of a maximal distribution, a member of the exponential family with nonlinear argument. We show that the entropy produced by heat transfer using a bosonic mode at weak coupling reproduces the maximal distribution in a limiting case. The upper bound is extended to the continuous domain and verified for the heat transfer using a levitated nanoparticle. Finally, we show that a composition of qubit swap engines satisfies a particular case of the maximal distribution regardless of its size.

quant-ph↗

arXiv2020

Non-linear Onsager relations for Gaussian quantum maps

Onsager's relations allow one to express the second law of thermodynamics in terms of the underlying associated currents. These relations, however, are usually valid only close to equilibrium. Using a quantum phase space formulation of the second law, we show that open bosonic Gaussian systems also obey a set of Onsager relations, valid arbitrarily far from equilibrium. These relations, however, are found to be given by a more complex non-linear function, which reduces to the usual quadratic form close to equilibrium. This non-linearity implies that far from equilibrium, there exists a fundamental asymmetry between entropy flow from system to bath and vice-versa. The ramifications of this for applications in driven-dissipative quantum optical setups are also discussed.

quant-ph↗

arXiv2019

Work distribution in thermal processes

We find the moment generating function (mgf) of the nonequilibrium work for open systems undergoing a thermal process, ie, when the stochastic dynamics maps thermal states into time dependent thermal states. The mgf is given in terms of a temperature-like scalar satisfying a first order ODE. We apply the result to some paradigmatic situations: a levitated nanoparticle in a breathing optical trap, a brownian particle in a box with a moving piston and a two state system driven by an external field, where the work mgfs are obtained for different timescales and compared with Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2019

Stochastic thermodynamics of nonharmonic oscillators in high vacuum

We perform an analytic study on the stochastic thermodynamics of a small classical particle trapped in a time dependent single-well potential in the highly underdamped limit. It is shown that the nonequilibrium probability density function for the system's energy is a Maxwell-Boltzmann distribution (as in equilibrium) with a closed form time dependent effective temperature and fractional degrees of freedom. We also find that the solvable model satisfies the Crooks fluctuation theorem, as it is expected. Moreover, we compute the average work in this isothermal process and characterize analytically the optimal protocol for minimum work. The optimal protocol presents an initial and a final jumps which correspond to adiabatic processes linked by a smooth exponential time dependent part for all kinds of single-well potentials. Furthermore, we argue that this result connects two distinct relevant experimental setups for trapped nano-particles: the levitated particle in a harmonic trap, and the free particle in a box; as they are limiting cases of the general single-well potential and display the time-dependent optimal protocols. Finally, we highlight the connection between our system and an equivalent model of a gas of Brownian particles.

cond-mat.stat-mech↗

arXiv2017

A maximum entropy approach to H-theory: Statistical mechanics of hierarchical systems

A novel formalism, called H-theory, is applied to the problem of statistical equilibrium of a hierarchical complex system with multiple time and length scales. In this approach, the system is formally treated as being composed of a small subsystem---representing the region where the measurements are made---in contact with a set of `nested heat reservoirs' corresponding to the hierarchical structure of the system. The probability distribution function (pdf) of the fluctuating temperatures at each reservoir, conditioned on the temperature of the reservoir above it, is determined from a maximum entropy principle subject to appropriate constraints that describe the thermal equilibrium properties of the system. The marginal temperature distribution of the innermost reservoir is obtained by integrating over the conditional distributions of all larger scales, and the resulting pdf is written in analytical form in terms of certain special transcendental functions, known as the Fox $H$-functions. The distribution of states of the small subsystem is then computed by averaging the quasi-equilibrium Boltzmann distribution over the temperature of the innermost reservoir. This distribution can also be written in terms of $H$-functions. The general family of distributions reported here recovers, as particular cases, the stationary distributions recently obtained by Macêdo {\it et al.} [Phys.~Rev.~E {\bf 95}, 032315 (2017)] from a stochastic dynamical approach to the problem.

cond-mat.stat-mech↗

arXiv2017

Nonequilibrium Thermodynamics of Restricted Boltzmann Machines

In this work, we analyze the nonequilibrium thermodynamics of a class of neural networks known as Restricted Boltzmann Machines (RBMs) in the context of unsupervised learning. We show how the network is described as a discrete Markov process and how the detailed balance condition and the Maxwell-Boltzmann equilibrium distribution are sufficient conditions for a complete thermodynamics description, including nonequilibrium fluctuation theorems. Numerical simulations in a fully trained RBM are performed and the heat exchange fluctuation theorem is verified with excellent agreement to the theory. We observe how the contrastive divergence functional, mostly used in unsupervised learning of RBMs, is closely related to nonequilibrium thermodynamic quantities. We also use the framework to interpret the estimation of the partition function of RBMs with the Annealed Importance Sampling method from a thermodynamics standpoint. Finally, we argue that unsupervised learning of RBMs is equivalent to a work protocol in a system driven by the laws of thermodynamics in the absence of labeled data.

cond-mat.stat-mech↗

arXiv2016

Exactly solvable nonequilibrium Langevin relaxation of a trapped nanoparticle

In this work, we study the nonequilibrium statistical properties of the relaxation dynamics of a nanoparticle trapped in a harmonic potential. We report an exact time-dependent analytical solution to the Langevin dynamics that arises from the stochastic differential equation of our system's energy in the underdamped regime. By utilizing this stochastic thermodynamics approach, we are able to completely describe the heat exchange process between the nanoparticle and the surrounding environment. As an important consequence of our results, we observe the validity of the heat exchange fluctuation theorem (XFT) in our setup, which holds for systems arbitrarily far from equilibrium conditions. By extending our results for the case of $N$ noninterating nanoparticles, we perform analytical asymptotic limits and direct numerical simulations that corroborate our analytical predictions.

cond-mat.stat-mech↗

arXiv2012

Multicanonical distribution: Statistical equilibrium of multiscale systems

A multicanonical formalism is introduced to describe statistical equilibrium of complex systems exhibiting a hierarchy of time and length scales, where the hierarchical structure is described as a set of nested "internal heat reservoirs" with fluctuating "temperatures." The probability distribution of states at small scales is written as an appropriate averaging of the large-scale distribution (the Boltzmann-Gibbs distribution) over these effective internal degrees of freedom. For a large class of systems the multicanonical distribution is given explicitly in terms of generalized hypergeometric functions. As a concrete example, it is shown that generalized hypergeometric distributions describe remarkably well the statistics of acceleration measurements in Lagrangian turbulence.

cond-mat.stat-mech↗

arXiv2010

Stochastic Dynamical Model of Intermittency in Fully Developed Turbulence

A novel model of intermittency is presented in which the dynamics of the rates of energy transfer between successive steps in the energy cascade is described by a hierarchy of stochastic differential equations. The probability distribution of velocity increments is calculated explicitly and expressed in terms of generalized hypergeometric functions of the type ${_n}F_0$, which exhibit power-law tails. The model predictions are found to be in good agreement with experiments on a low temperature gaseous helium jet. It is argued that distributions based on the functions ${_n}F_0$ might be relevant also for other physical systems with multiscale dynamics.

physics.flu-dyn↗