Brownian motion of particle interacting with atoms of ideal gas is discussed as a key problem of kinetics lying at the border between ``dead'' systems like the Lorentz gas or formal constructs of conceptual Boltzmannian kinetics and actual ``alive'' systems like mere gas possessing scaleless (1/f) fluctuations in their kinetic characteristics (e.g. in diffusuvity and mobility of the ``Brownian particle'').
An attempt is made to compare statistical properties of self-diffusion of particles constituting gases in infinite volume and on torus. In this first part, equations are derived which represent roughened but solvable variant of the collisional approximation to exact BBGKY equations. With their help, statistics of Brownian motion in infinite gas is considered, under the Boltzmann-Grad limit, and shown to be essentially non-Gaussian, involving 1/f fluctuations in diffusivity.
A Brownian particle in an ideal quantum gas is considered. The mean square displacement (MSD) is derived. The Bose-Einstein or Fermi-Dirac distribution, other than the Maxwell-Boltzmann distribution, provides a different stochastic force compared with the classical Brownian motion. The MSD, which depends on the thermal wavelength and the density of medium particles, reflects the quantum effect on the Brownian particle explicitly. The result shows that the MSD in an ideal Bose gas is shorter than that in a Fermi gas. The behavior of the quantum Brownian particle recovers the classical Brownian …
An attempt is made to compare statistical properties of self-diffusion of particles constituting gases in infinite volume and on torus. In this second part, derivation, from BBGKY equations, of roughened model of self-diffusion is revised as applied to finite $N$-particle gas under micro-canonical ensemble. The model confirms existence of characteristic time $\approx N$, in units of free flight time, for cross-over between non-Gaussian and Gaussian regimes of diffusion, but then loses its legacy.
We calculate the systematic average speed of the adiabatic piston and a thermal Brownian motor, introduced in [Van den Broeck, Kawai and Meurs, \emph{Microscopic analysis of a thermal Brownian motor}, to appear in Phys. Rev. Lett.], by an expansion of the Boltzmann equation and compare with the exact numerical solution.
A formally exact relation is derived which connects thermodynamically non-equilibrium evolution of gas density distribution after its arbitrary strong spatially non-uniform perturbation and evolution of many-particle correlations between path of some marked particle and its surroundings in equilibrium gas. This relation directly confirms significance of the many-particle correlations even under the Boltzmann-Grad limit and thus validates the earlier suggested revision of kinetics.
We develop a formally exact technique for obtaining steady-state distributions of non-interacting active Brownian particles in a variety of systems. Our technique draws on results from the theory of two-way diffusion equations to solve the steady-state Smoluchowski equation for the 1-particle distribution function. The methods are employed to study in detail three scenarios: 1) confinement in a channel, 2) a constant flux steady state, and 3) sedimentation in a uniform external field. In each scenario, known behaviors are reproduced and precisely quantified, and new results are presented. In p…
The extent to which active matter might be described by effective equilibrium concepts like temperature and pressure is currently being discussed intensely. Here we study the simplest model, an ideal gas of non-interacting active Brownian particles. While the mechanical pressure exerted onto confining walls has been linked to correlations between particles' positions and their orientations, we show that these correlations are entirely controlled by boundary effects. We also consider a definition of local pressure, which describes interparticle forces in terms of momentum exchange between diffe…
Explicit density expansions of non-equilibrium probability distribution functions for molecular Brownian particle in ideal gas are obtained in original form what visually implies (is exact solution to) the previously established dynamical virial relations. Role of these relations in unbiased analysis of molecular chaos properties in many-particle statistical mechanics, including the mobility 1/f noise, is newly investigated in clear terms of conditional probabilities and averages.
In contrast with a gas at thermodynamic equilibrium, the mean force exerted on a wall by a gas of active particles usually depends on the confining potential, thereby preventing a proper definition of mechanical pressure. In this paper, we investigate numerically the properties of a gas of underdamped self-propelled dumbbells subject to Brownian noise of increasing intensity, in order to understand how the notion of pressure is recovered as noise progressively masks the effects of self-propulsion and the system approaches thermodynamic equilibrium. The simulations performed for a mobile asymme…
A model of an autonomous isothermal Brownian motor with an internal propulsion mechanism is considered. The motor is a Brownian particle which is semi-transparent for molecules of surrounding ideal gas. Molecular passage through the particle is controlled by a potential similar to that in the transition rate theory, i.e. characterized by two stationary states with a finite energy difference separated by a potential barrier. The internal potential drop maintains the diode-like asymmetry of molecular fluxes through the particle, which results in the particle's stationary drift.
A covariant Fokker-Planck type equation for a simple gas and an equation for the Brownian motion are derived from a relativistic kinetic theory based on the Boltzmann equation. For the simple gas the dynamic friction four-vector and the diffusion tensor are identified and written in terms of integrals which take into account the collision processes. In the case of Brownian motion, the Brownian particles are considered as non-relativistic whereas the background gas behaves as a relativistic gas. A general expression for the semi-relativistic viscous friction coefficient is obtained and the part…
We analyze the dynamics of a Brownian gas in contact with a heat bath in which large temperature fluctuations occur. There are two distinct time scales present, one describes the decay of the fluctuations in the temperature and the other one is associated with the establishment of local equilibrium. Although the gas has reached local equilibrium, there exist large fluctuations in an intensive parameter (temperature) which break the thermodynamic equilibrium with the heat bath. Thus the decay of the fluctuations in the intensive parameter is larger than the characteristic time for the establish…
The BBGKY hierarchy of equations for a particle interacting with ideal gas is analyzed in terms of irreducible many-particle correlations between gas atoms and the particle's motion. The transition to the hard-sphere interaction is formulated from viewpoint of the recently discovered exact relations connecting the correlations with the particle's probability distribution. Then the Boltzmann-Grad limit is considered and shown not to lead to the Bolzmann hierarchy and the molecular chaos, since correlations of all orders keep significant.in this limit, merely taking a singular form.
We investigate the motion of a wedge-shaped object (a granular Brownian motor), which is restricted to move along the x-axis and cannot rotate, as gas particles collide with it. We show that its steady-state drift, resulting from inelastic gas-motor collisions, is dramatically affected by anisotropy in the velocity distribution of the gas. We identify the dimensionless parameter providing the dependence of this drift on shape, masses, inelasticity, and anisotropy: the anisotropy leads to a dramatic breaking of equipartition, which should easily be visible in experimental realizations.
Real thermal motion of gas molecules, free electrons, etc., at long time intervals (much greater than mean free-flight time) possesses, contrary to its popular mathematical models, essentially non-Gaussian statistics. A simple proof of this statement is suggested basing on only the determinism and reversibility of microscopic dynamics and besides incidentally derived virial expansion of a path probability distribution of molecular Brownian particle.
We present an insightful ``derivation'' of the Langevin equation and the fluctuation dissipation theorem in the specific context of a heavier particle moving through an ideal gas of much lighter particles. The Newton's Law of motion ($m{\ddot x}=F$) for the heavy particle reduces to a Langevin equation (valid on a coarser time scale) with the assumption that the lighter gas particles follow a Boltzmann velocity distribution. Starting from the kinematics of the random collisions we show that (1) the average force $<F> \propto -{\dot x}$ and (2) the correlation function of the fluctuating force …
We investigate a model describing the dynamics of a gas of self-gravitating Brownian particles. This model can also have applications for the chemotaxis of bacterial populations. We focus here on the collapse phase obtained at sufficiently low temperature/energy and on the post-collapse regime following the singular time where the central density diverges. Several analytical results are illustrated by numerical simulations.
Brownian motion in a granular gas in a homogeneous cooling state is studied theoretically and by means of molecular dynamics. We use the simplest first-principle model for the impact-velocity dependent restitution coefficient, as it follows for the model of viscoelastic spheres. We reveal that for a wide range of initial conditions the ratio of granular temperatures of Brownian and bath particles demonstrates complicated non-monotonous behavior, which results in transition between different regimes of Brownian dynamics: It starts from the ballistic motion, switches later to superballistic one …
Motility-induced phase separation (MIPS), the phenomenon in which purely repulsive active particles undergo a liquid-gas phase separation, is among the simplest and most widely studied examples of a nonequilibrium phase transition. Here, we show that states of MIPS coexistence are in fact only metastable for three-dimensional active Brownian particles over a very broad range of conditions, decaying at long times through an ordering transition we call active crystallization. At an activity just above the MIPS critical point, the liquid-gas binodal is superseded by the crystal-fluid coexistence …
The evolution of prices on ideal market is given by geometrical Brownian motion, where Gaussian white noise describes fluctuations. We study the effect of correlations introduced by a color noise.
Exact generalized stochastic representation of deterministic interaction between two dynamical (quantum or classical) systems is derived which helps when considering one of them to replace another by equivalent commutative ($c$-number valued) random sources. The method is applied to classical Brownian motion of a particle in a gas, and statistics of this motion is reduced to statistics of the gas response to perturbations.
We derive a dynamic Ornstein-Zernike equation for classical fluids undergoing overdamped Brownian motion and driven out of equilibrium. Inhomogeneous two-time correlation functions are obtained from functional differentiation of the one-body density and current with respect to an appropriately chosen external field. Functional calculus leads naturally to non-Markovian equations of motion for the two-time correlators. Memory functions are identified as functional derivatives of a space- and time-nonlocal power dissipation functional. We propose an excess (over ideal gas) dissipation functional …
We investigate the fluctuations of the condensate in the ideal and weakly interacting Bose gases confined in a box of volume V within canonical ensemble. Canonical ensemble is developed to describe the behavior of the fluctuations when different methods of approximation to the weakly interacting Bose gases are used. Research shows that the fluctuations of the condensate exhibit anomalous behavior for the interacting Bose gas confined in a box.
Basing on main principles of statistical mechanics only, an exact virial expansion for path probability distribution of molecular Brownian particle in a fluid is derived which connects response of the distribution to perturbations of the fluid and statistical correlations of its molecules with Brownian particle. The expansion implies that (i) spatial spread of these correlations is finite, (ii) this is inconsistent with Gaussian distribution involved by the ``molecular chaos'' hypothesis, and (iii) real path distribution possesses power-law long tails. This means that actual Brownian path neve…
We present a microscopic theory for the nonequilibrium interfacial tension $γ_{\rm gl}$ of the free interface between gas and liquid phases of active Brownian particles. The underlying square gradient treatment and the splitting of the force balance in flow and structural contributions is general and applies to inhomogeneous nonequilibrium steady states. We find $γ_{\rm gl}\geq 0$, which opposes claims by Bialké et al. [Phys. Rev. Lett. $\bf{115}$, 098301 (2015)] and delivers the theoretical justification for the widely observed interfacial stability in active Brownian dynamics many-body simul…
A theory of Brownian motion is presented for an assembly of vortices. The attempt is motivated by a realization of Dyson' Coulomb gas in the context of quantum condensates. By starting with the time-dependent Landau-Ginzburg (LG) theory, the dynamics of the vortex gas is constructed, which is governed by the canonical equation of motion. The dynamics of point vortices is converted to the Langevin equation, which results in the generalized Fokker-Planck (GFP) (or Smolkovski) equation using the functional integral on the ansatz of the Gaussian white noise. The GFP, which possesses a non-Hermitia…
We consider a gas of independent Brownian particles on a bounded interval in contact with two particle reservoirs at the endpoints. Due to the Brownian nature of the particles, infinitely many particles enter and leave the system in each time interval. Nonetheless, the dynamics can be constructed as a Markov process with continuous paths on a suitable space. If $λ_0$ and $λ_1$ are the chemical potentials of the boundary reservoirs, the stationary distribution (reversible if and only if $λ_0=λ_1$) is a Poisson point process with intensity given by the linear interpolation between $λ_0$ and $λ_1…
We consider the dynamics of a gas of free bosons within a semi-classical Fokker-Planck equation for which we give a physical justification. In this context, we find a striking similarity between the Bose-Einstein condensation in the canonical ensemble, and the gravitational collapse of a gas of classical self-gravitating Brownian particles. The paper is mainly devoted to the complete study of the Bose-Einstein ``collapse'' within this model. We find that at the Bose-Einstein condensation temperature $T_c$, the chemical potential $μ(t)$ vanishes exponentially with a universal rate that we compu…
Einstein realised that the fluctuations of a Brownian particle can be used to ascertain properties of its environment. A large number of experiments have since exploited the Brownian motion of colloidal particles for studies of dissipative processes, providing insight into soft matter physics, and leading to applications from energy harvesting to medical imaging. Here we use optically levitated nanospheres that are heated to investigate the non-equilibrium properties of the gas surrounding them. Analysing the sphere's Brownian motion allows us to determine the temperature of the centre-of-mass…
As a first step towards a successful field theory of Brownian particles in interaction, we study exactly the non-interacting case, its combinatorics and its non-linear time-reversal symmetry. Even though the particles do not interact, the field theory contains an interaction term: the vertex is the hallmark of the original particle nature of the gas and it enforces the constraint of a strictly positive density field, as opposed to a Gaussian free field. We compute exactly all the n-point density correlation functions, determine non-perturbatively the Poissonian nature of the ground state and e…
In the air surrounding us, how does a particle diffuse? Thanks to Einstein and other pioneers,it has been well known that generally the particle will undergo the Brownian motion, and in the last century this insight has been corroborated by numerous experiments and applications. Another fundamental question is how the energy carried by a particle diffuses. The conventional transport theories assumed the Brownian motion as the underlying energy transporting mechanism, but however, it should be noticed that in fact this assumption has never been tested and verified directly in experiments. Here …
We present analytical formulae for the first and second derivatives of the Helmholtz free energy of non-relativistic ideal Fermi-gas. Important thermodynamic quantities such as heat capacity, sound velocity, heat capacity ratio and others can be easily expressed through the derivatives. We demonstrate correct ideal Boltzmann gas and low--temperature Fermi-gas asymptotes and derive corrections to thermodynamic functions for these limiting cases. Numerical computations of thermodynamic properties of ideal Fermi-gas can be accurately performed using the developed freely available Python module if…
The equation of state for a two-dimensional hard-sphere gas is difficult to calculate by usual methods. In this paper we develop an approach for calculating the equation of state of hard-sphere gases, both for two- and three-dimensional cases. By regarding a hard-sphere gas as an ideal gas confined in a container with a multi-core (excluded sphere) boundary, we treat the hard-sphere interaction in an interacting gas as the boundary effect on an ideal quantum gas; this enables us to treat an interacting gas as an ideal one. We calculate the equation of state for a three-dimensional hard-sphere …
An english translation of the original work of Enrico Fermi on the quantization of the monoatomic ideal gas, is given in this paper. Fermi introduced here for the first time the statistical law obeyed by most of the particles in nature: Fermi-Dirac statistics. The formalism that he used is based on the quantization of the monoatomic ideal gas in a harmonic trap. This trappping configuration is similar to that used in current experiments on trapped Fermi gases.
We explore the relation between active Brownian particles, a minimal particle-based model for active matter, and scalar field theories. Both show a liquid-gas-like phase transition towards stable coexistence of a dense liquid with a dilute active gas that terminates in a critical point. However, a comprehensive mapping between the particle-based model parameters and the effective coefficients governing the field theories has not been established yet. We discuss conflicting recent numerical results for the critical exponents of active Brownian particles in two dimensions. Starting from the inte…
We consider a charged Brownian gas under the influence of external and non uniform electric, magnetic and mechanical fields, immersed in a non uniform bath temperature. With the collision time as an expansion parameter, we study the solution to the associated Kramers equation, including a linear reactive term. To first order we obtain the asymptotic (overdamped) regime, governed by transport equations, namely: for the particle's density, a Smoluchowski-reactive like equation; for the particle's momentum density, a generalized Ohm's like equation; and for the particle's energy density, a Maxwel…
We develop the analogy between self-gravitating Brownian particles and bacterial populations. In the high friction limit, the self-gravitating Brownian gas is described by the Smoluchowski-Poisson system. These equations can develop a self-similar collapse leading to a finite time singularity. Coincidentally, the Smoluchowski-Poisson system corresponds to a simplified version of the Keller-Segel model of bacterial populations. In this biological context, it describes the chemotactic aggregation of the bacterial colonies. We extend these classical models by introducing a small-scale regularizat…
The probability distribution of the entropy production for the effusion of a relativistic ideal gas is calculated explicitly. This result is then extended to include particle and anti-particle pair production and annihilation. In both cases, the fluctuation theorem is verified.
Based on the prescription termed the optimal Lagrange multipliers formalism for extremizing the Tsallis entropy indexed by q, it is shown that key aspects of the treatment of the ideal gas problem are identical in both the nonextensive and extensive cases.
The equations of state for an ideal generalized gas, like an ideal quantum gas, are expressed in terms of power laws of the temperature. The reduction of an ideal generalized gas to an ideal classical case occurs when the characteristic empirical temperature exponents in the thermal equation of state and in the absolute temperature coincide in contrast to the merger of an ideal quantum gas with an ideal classical gas in the high temperature limit. A corollary to Carnot's theorem is proved asserting that the ratio of the work done over a cycle to the heat absorbed to increase the temperature at…
The probability distribution of the entropy production for the effusion of an ideal gas between two compartments is calculated explicitly. The fluctuation theorem is verified. The analytic results are in good agreement with numerical data from hard disk molecular dynamics simulations.
The Fokker-Planck equation for a heavy particle in a granular fluid is derived from the Liouville equation. The host fluid is assumed to be in its homogeneous cooling state and all interactions are idealized as smooth, inelastic hard spheres. The similarities and differences between the Fokker-Planck equation for elastic and inelastic collisions are discussed in detail. Although the fluctuation-dissipation relation is violated and the reference fluid is time dependent, it is shown that diffusion occurs at long times for a wide class of initial conditions. The results presented here generalize …
We investigate theoretically the properties of an ideal trapped gas in a time-dependent harmonic potential. Using a scaling formalism, we are able to present simple analytical results for two important classes of experiments: free expansion of the gas upon release of the trap; and the response of the gas to a harmonic modulation of the trapping potential is investigated. We present specific results relevant to current experiments on trapped Fermions.
Landauer principle states that erasure of $N$ bit information requires an entropic cost of $Nk_B \ln 2$. This fact can easily be demonstrated by relaxation of an ideal gas consisting of $N$ particles inside a fixed volume. In this paper we discuss the modification of Landauer bound when we use non-ideal gas with inter-particle interactions for erasure procedure. We have found that the bound on the entropy production can be lowered when the interaction between particles is square-well potential.
We study the thermodynamical properties of a self-gravitating gas with two or more types of particles. Using the method of linear series of equilibria, we determine the structure and stability of statistical equilibrium states in both microcanonical and canonical ensembles. We show how the critical temperature (Jeans instability) and the critical energy (Antonov instability) depend on the relative mass of the particles and on the dimension of space. We then study the dynamical evolution of a multi-components gas of self-gravitating Brownian particles in the canonical ensemble. Self-similar sol…
A new type of Coulomb gas is defined, consisting of arbitrary numbers of point charges of two species executing Brownian motions under the influence of their mutual electrostatic repulsion. Being a generalization of a model of identical particles introduced by Dyson as a dynamical system describing non-equilibrium state of various random matrix ensembles, our system gives an exact mathematical description of the Brownian motion of charges of magnitudes $ Q_1 $ and $ Q_2 $, such that $ βQ_1 Q_2 = 1 $ on the line or circle, where $β$ is an inverse temperature of the gas.
We consider both Otto and Diesel heat engine cycles running upon the working substances modeled by the van der Waals fluid as a simple non-ideal gas model. We extensively perform the efficiency study in these model engines. Then we find that this "real" engine model can go beyond its ideal-gas counterpart in efficiency, whereas as well-known, the maximum Carnot efficiency is the same for both ideal and non-ideal gas engines. In fact, the higher the density of non-ideality is, the higher efficiency tends to be found, especially in the low-temperature regime, but with more shrinkage in the range…
We consider a heavy piston in an infinite cylinder surrounded by ideal gases on both sides. The piston moves under elastic collisions with gas atoms. We assume here that the gases always exert equal pressures on the piston, hence the piston remains at the so called mechanical equilibrium. However, the temperatures and densities of the gases may differ across the piston. In that case some earlier studies by Gruber, Piasecki and others reveal a very slow motion (drift) of the piston in the direction of the hotter gas. At the same time the hotter gas slowly transfers its energy (heat) across the …
The virial expansion of a gas is a correction to the ideal gas law that is usually discussed in advanced courses in statistical mechanics. In this note we outline this derivation in a manner suitable for advanced undergraduate and introductory graduate classroom presentations. We introduce a physically meaningful interpretation of the virial expansion that has heretofore escaped attention, by showing that the virial series is actually an expansion in a parameter that is the ratio of the effective volume of a molecule to its mean volume. Using this interpretation we show why under normal condit…
We derive the exact expression of the diffusion coefficient of a self-gravitating Brownian gas in two dimensions. Our formula generalizes the usual Einstein relation for a free Brownian motion to the context of two-dimensional gravity. We show the existence of a critical temperature T_{c} at which the diffusion coefficient vanishes. For T<T_{c} the diffusion coefficient is negative and the gas undergoes gravitational collapse. This leads to the formation of a Dirac peak concentrating the whole mass in a finite time. We also stress that the critical temperature T_{c} is different from the colla…
We consider the thermodynamic geometry of an ideal non-Abelian gas. We show that, for a certain value of the fractional parameter and at the relevant maximum value of fugacity, the thermodynamic curvature has a singular point. This indicates a condensation such as Bose-Einstein condensation for non-Abelian statistics and we work out the phase transition temperature in various dimensions.
The ideal Bose-gas with finite number $N$ of particles is investigated. The exact expressions for the partition functions and occupation numbers in the grand canonical, canonical and microcanonical ensembles are found. The asymp\-totic expressions (in the case $N\gg1$) for the partition functions and occupation numbers in the canonical and microcanonical ensembles are evaluated. It is shown that the chemical potential $μ$ of the ideal Bose-gas can lie in the range $-\infty<μ<\infty$ oppositely to the widely adopted opinion that the value of this potential is negative.
We consider a charged Brownian gas under the influence of external, static and uniform electric and magnetic fields, immersed in a uniform bath temperature. We obtain the solution for the associated Langevin equation, and thereafter the evolution of the nonequilibrium temperature towards a nonequilibrium (hot) steady state. We apply our results to a simple yet relevant Brownian model for carrier transport in GaAs. We obtain a negative differential conductivity regime (Gunn effect) and discuss and compare our results with experimental results.
