arXiv subjects

Mário J. de Oliveira

Publications and source records attributed to Mário J. de Oliveira.

At least 19 recordsLinked to original sources

arXiv2026

Derivation of Fokker-Planck equation and its entropy production

We derive the Fokker-Planck equation and its entropy production from the master equation. To this end we consider a discrete space of states where the master equation is defined and appropriate transition rates are introduced. A transition rate has two parts, one of which is the ordinary term proportional to the inverse of the spacing and the other is inversely proportional to the square of the spacing corresponding to the variance of the independent stochastic variable. In the continuous space limit, we obtain the Fokker-Planck as well as the rate of entropy production and the flux of entropy. We also consider the case where the space of states is the phase space and discuss the equations appropriate for systems in contact with several heat reservoirs and for isolated systems.

cond-mat.stat-mech↗

arXiv2026

Gibbs variational principles and Boltzmann irreversible theorem

We analyze the Gibbs variational principles associated with the probability distributions of (i) an isolated system and (ii) a system at constant temperature. We give an example of using the Gibbs inequality to obtain the free energy and analyze the phase diagram of an Ising model with competing interactions. We also review the Boltzmann irreversible theorem, and show how it is connected to the Gibbs variational principles. This connection is established by using the Kolmogorov equation for the evolution of the probability distribution, which predicts a monotonic increase of entropy for an isolated system, and a decrease of the free energy for a system in contact with a thermal reservoir.

cond-mat.stat-mech↗

arXiv2026

Boltzmann-Kolmogorov equation

We investigate the properties of a Kolmogorov equation governing the time evolution of the probability distribution defined in phase space. Energy is strictly conserved along a trajectory in phase space, meaning the equation is appropriate to describe an isolated system, and the stationary state is the Gibbs microcanonical distribution. The equation predicts the increase in entropy in agreement with thermodynamics, and in contrast with the Liouville equation, which conserves entropy. Using an approximation in which the distribution is a product of one-particle distributions, we derive the Boltzmann equation of kinetic theory. We also consider a Kolmogorov equation to describe an open system in contact with the external environment. In this case the equation describes not only the situation in which the system is found in thermodynamic equilibrium with a Gibbs canonical distribution in the stationary state, but also the nonequilibrium steady state with a continuous production of entropy.

cond-mat.stat-mech↗

arXiv2026

Stochastic interpretation of quantum mechanics

We express the probabilistic character associated to the wave function by treating it as a stochastic variable. This is accomplished by means of a stochastic equation for the wave function whose noise changes the phase of the wave function but not its absolute value, so that the norm of the wave function is strictly conserved along a stochastic trajectory. We show that the density matrix that obeys the quantum Liouville equation is the covariance matrix associated to the stochastic wave function.

quant-ph↗

arXiv2025

Origins of the Ising model

In 1925, Ernest Ising published a paper analyzing a model proposed in 1920 by Wilhelm Lenz for ferromagnetism. The model is composed of constituent units that take only two states and interact only when they are neighbors. Ising showed that in a linear chain the model does not present an ordered ferromagnetic state, a frustrating but correct result. However, Rudolf Peierls demonstrated in 1936 that the model does in fact present an ordered state in two dimensions, and therefore in three dimensions. This result reveals that short-range interaction and only two states for each constituent unit are sufficient for ordering to occur over long distances. These two elements are the key to understanding the success of the model and its variants even a hundred years after its appearance. Here we analyze the emergence of the model in the period up to 1936.

physics.hist-ph↗

arXiv2025

Irreversible thermodynamics and Glansdorff-Prigogine principle derived from stochastic thermodynamics

We derive the main equations of irreversible thermodynamic including the expression for the Glansdorff-Prigogine extremal principle from stochastic thermodynamics. To this end, we analyze a system that is subject to gradients of temperature and external forces that induce the appearance of fluxes of several sorts and the creation of entropy. We show that the rate of entropy production is a convex function of the fluxes, from which follows that the excess entropy production is nonnegative, which is an expression of the Glansdorff-Prigogine principle. We show that the Lyapunov function associated with the excess entropy production can be identified with a thermodynamic potential in the special case where the gradients of temperature are absent.

cond-mat.stat-mech↗

arXiv2024

Entropy production along a deterministic motion

We propose a stochastic dynamics to be associated to a deterministic motion defined by a set of first order differential equation. The transitions that defined the stochastic dynamics are unidirectional and the rates are equal to the absolute value of the velocity vector field associate to the deterministic motion. From the stochastic dynamics we determine the entropy production and the entropy flux. This last quantity is found to be the negative of the divergence of the velocity vector field. In the case of a Hamiltonian dynamics it vanishes identically.

cond-mat.stat-mech↗

arXiv2024

Entropy production in continuous systems with unidirectional transitions

We derive the expression for the entropy production for stochastic dynamics defined on a continuous space of states containing unidirectional transitions. The expression is derived by taking the continuous limit of a stochastic dynamics on a discrete space of states and is based on an expression for the entropy production appropriate for unidirectional transition. Our results shows that the entropy flux is the negative of the divergence of the vector firld whose components are the rates at which a dynamic variable changes in time. For a Hamiltonian dynamical system, it follows from this result that the entropy flux vanish identically.

cond-mat.stat-mech↗

arXiv2024

Entropy production of the contact model

We propose an expression for the production of entropy for system described by a stochastic dynamics which is appropriate for the case where the reverse transition rate vanishes but the forward transition is nonzero. The expression is positive defined and based on the inequality $x\ln x-(x-1)\ge0$. The corresponding entropy flux is linear in the probability distribution allowing its calculation as an average. The expression is applied to the one-dimensional contact process at the stationary state. We found that the rate of entropy production per site is finite with a singularity at the critical point with diverging slope.

cond-mat.stat-mech↗

arXiv2024

Stochastic thermodynamics of ecosystems

We investigate the thermodynamics as well as the population dynamics of ecosystems based on a stochastic approach in which the number of individuals of the several species of the ecosystem are treated as stochastic variables. The several species are connected by feeding relationships that are understood as unidirectional processes in which a certain amount of biomass is exchanged between species. We show that the equations for the averages in the numbers of individuals are that given by the deterministic approach. We determine the fluxes of mass, energy, and entropy as well as the rate of the entropy production. This last quantity, which has a central role in the present stochastic approach, is obtained by a formula appropriate for unidirectional transitions. The flux of energy across the ecosystem is shown to be proportional to the flux of entropy to the environment.

cond-mat.stat-mech↗

arXiv2023

Derivation of the Schrödinger equation from classical stochastic dynamics

From classical stochastic equations of motion we derive the quantum Schrödinger equation. The derivation is carried out by assuming that the real and imaginary parts of the wave function $ϕ$ are proportional to the coordinates and momenta associated to the degrees of freedom of an underlying classical system. The wave function $ϕ$ is assumed to be a complex time dependent random variable that obeys a stochastic equation of motion that preserves the norm of $ϕ$. The quantum Liouville equation is obtained by considering that the stochastic part of the equation of motion changes the phase of $ϕ$ but not its absolute value. The Schrödinger equation follows from the Liouville equation. The wave function $ψ$ obeying the Schrödinger equation is related to the stochastic wave function by $|ψ|^2=\langle|ϕ|^2\rangle$.

quant-ph↗

arXiv2023

Nonequilibrium thermodynamics of the majority vote model

The majority vote model is one of the simplest opinion systems yielding distinct phase transitions and has garnered significant interest in recent years. However, its original formulation is not, in general, thermodynamically consistent, precluding the achievement of quantities such as power and heat, as well as their behaviors at phase transition regimes. Here, we circumvent this limitation by introducing the idea of a distinct heat bath per local configuration, in such a way that each neighborhood value is associated with a distinct and well-defined thermal bath. Thermodynamic properties are derived for a generic majority vote model, irrespective of its neighborhood and lattice topology. The behavior of energy/heat fluxes at phase transitions, whether continuous or discontinuous, in regular and complex topologies, is investigated in detail. Unraveling the contribution of each local configuration explains the nature of the phase diagram and reveals how dissipation arises from the dynamics.

cond-mat.stat-mech↗

arXiv2023

Quantum Fokker-Planck structure of the Lindblad equation

We show that the quantum Fokker-Planck equation, obtained by a canonical quantization of its classical version, can be transformed into an equation of the Lindblad form. This result allows us to conclude that the quantum Fokker-Planck equation preserves the trace and positivity of the density operator. The Fokker-Planck structure gives explicit expression for the quantum equivalence of probability current as well as the quantum equivalence of detailed balance. We also propose expression for the rate of entropy production and show that it does not vanish for a closed system except in equilibrium.

cond-mat.stat-mech↗

arXiv2022

Stochastic thermodynamics of opinion dynamics

We show that models of opinion formation and dissemination in a community of individuals can be framed within stochastic thermodynamics from which we can build a nonequilibrium thermodynamics of opinion dynamics. This is accomplished by decomposing the original transition rate that defines an opinion model into two or more transition rates, each representing the contact with heat reservoirs at different temperatures, and postulating an energy function. As the temperatures are distinct, heat fluxes are present even at the stationary state and linked to the production of entropy, the fundamental quantity that characterizes nonequilibrium states. We apply the present framework to a generic-vote model including the majority-vote model in a square lattice and in a cubic lattice. The fluxes and the rate of entropy production are calculated by numerical simulation and by the use of a pair approximation.

cond-mat.stat-mech↗

arXiv2022

Stochastic motion in phase space on a surface of constant energy

We study closed systems of particles that are subject to stochastic forces in addition to the conservative forces. The stochastic equations of motion are set up in such a way that the energy is strictly conserved at all times. To ensure this conservation law, the evolution equation for the probability density is derived using an appropriate interpretation of the stochastic equation of motion that is not the Itô nor the Stratonovic interpretation. The trajectories in phase space are restricted to the surface of constant energy. Despite this restriction, the entropy is shown to increase with time, expressing irreversible behavior and relaxation to equilibrium. This main result of the present approach contrasts with that given by the Liouville equation, which also describes closed systems, but does not show irreversibility.

cond-mat.stat-mech↗

arXiv2022

Parametric invariance

We examine the development of the concept of parametric invariance in classical mechanics, quantum mechanics, statistical mechanics, and thermodynamics, and particularly its relation to entropy. The parametric invariance was used by Ehrenfest as a principle related to the quantization rules of the old quantum mechanics. It was also considered by Rayleigh in the determination of pressure caused by vibration, and the general approach we follow here is based on his. Specific calculation of invariants in classical and quantum mechanics are determined. The Hertz invariant, which is a volume in phase space, is extended to the case of a variable number of particles. We show that the slow parametric change leads to the adiabatic process, allowing the definition of entropy as a parametric invariance.

cond-mat.stat-mech↗

arXiv2021

Effect of immunization through vaccination on the SIS epidemic spreading model

We analyze the susceptible-infected-susceptible model for epidemic spreading in which a fraction of the individuals become immune by vaccination. This process is understood as a dilution by vaccination, which decreases the fraction of the susceptible individuals. For a nonzero fraction of vaccinated individuals, the model predicts a new state in which the disease spreads but eventually becomes extinct. The new state emerges when the fraction of vaccinated individuals is greater than a critical value. The model predicts that this critical value increases as one increases the infection rate reaching an asymptotic value, which is strictly less than the unity. Above this asymptotic value, the extinction occurs no matter how large the infection rate is.

q-bio.PE↗

arXiv2020

Epidemic spreading

We present an analysis of six deterministic models for epidemic spreading. The evolution of the number of individuals of each class is given by ordinary differential equations of the first order in time, which are set up by using the laws of mass action providing the rates of the several processes that define each model. The epidemic spreading is characterized by the frequency of new cases, which is the number of individuals that are becoming infected per unit time. It is also characterized by the basic reproduction number, which we show to be related to the largest eigenvalue of the stability matrix associated with the disease-free solution of the evolution equations. We also emphasize the analogy between the outbreak of an epidemic with a critical phase transition. When the density of the population reaches a critical value the spreading sets in, a result that was advanced by Kermack and McKendrick in their study of a model in which the recovered individuals acquire permanent immunization, which is one of the models analyzed here.

physics.bio-ph↗