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 particle as the temperature raises. At low temperatures, the quantum effect becomes obvious. For example, there is a random motion of the Brownian particle due to the fermionic exchange interaction even the temperature is near the absolute zero.
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'').
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.
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 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 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.
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 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.
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…
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 …
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.
We investigate the motility of a growing population of cells in a idealized setting: we consider a system of hard disks in which new particles are added according to prescribed growth kinetics, thereby dynamically changing the number density. As a result, the expected Brownian motion of the hard disks is modified. We compute the density-dependent friction of the hard disks and insert it in an effective Langevin equation to describe the system, assuming that the inter-collision time is smaller than the timescale of the growth. We find that the effective Langevin description captures the changes…
A pseudo-velocity concept, based on the extension of a linear body, is defined by a special relativity experiment. It suggests an analogy with the covariance properties of Wiener's process, ultimately implying that the scaling behavior of (Gaussian) polymer solutions can be derived from a Brownian Relativity theory, as it was formerly put forward. An ad-hoc statistical interpretation of the resulting spacetime transforms may be given by the central limit theorem.
We study the motility-induced phase separation of active particles driven through the interconversion of two chemical species controlled by ideal reservoirs (chemiostats). As a consequence, the propulsion speed is non-constant and depends on the actual inter-particle forces, enhancing the positive feedback between increased density and reduced motility that is responsible for the observed inhomogeneous density. For hard discs, we find that this effect is negligible and that the phase separation is controlled by the average propulsion speed. For soft particles and large propulsion speeds, howev…
We investigate how to coherently define entropy production for a process of transient relaxation in the Quantum Brownian Motion model for harmonic potential. We compare a form, called "Poised" (P), which after non-Markovian transients corresponds to a definition of heat as the change in the system Hamiltonian of mean force, with a recent proposal by Esposito (ELB) based on a definition of heat as the energy change in the bath. Both expressions yield a positive-defined entropy production and coincide for vanishing coupling strength, but their difference is proved to be always positive (after no…
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 …
The paper contains a simple semi-quantitative analysis of a structure of solution to the exact Bogolyubov functional equation for a particle interacting with ideal gas and driven by an external force, in comparison with solutions to model kinetic equations for the same system. It is shown that the exact equation inevitably predicts existence of significant 1/f-type fluctuations in mobility of the particle, and this result directly extends to particles in arbitrary fluid.
Interest in Brownian motion was shared by different communities: this phenomenon was first observed by the botanist Robert Brown in 1827, then theorised by physicists in the 1900s, and eventually modelled by mathematicians from the 1920s, while still evolving as a physical theory. Consequently, Brownian motion now refers to the natural phenomenon but also to the theories accounting for it. There is no published work telling its entire history from its discovery until today, but rather partial histories either from 1827 to Perrin's experiments in the late 1900s, from a physicist's point of view…
Statistics of classical Hamiltonian random walk of particle colliding with atoms of ideal gas is considered from viewpoint of earlier suggested exact pseudo-quantum path integral representation of the problem, and qualitative agreement is demostrated between results of an naturally arising simple approximation of this integral and results obtained by formally different methods, thus in a new fashion showing inevitability of scaleless 1/f-type fluctuations in rates of molecular Brownian motions and other dynamical transport and evolution processes.
The motion of particles in random potentials occurs in several natural phenomena ranging from the mobility of organelles within a biological cell to the diffusion of stars within a galaxy. A Brownian particle moving in the random optical potential associated to a speckle, i.e., a complex interference pattern generated by the scattering of coherent light by a random medium, provides an ideal mesoscopic model system to study such phenomena. Here, we derive a theory for the motion of a Brownian particle in a speckle and, in particular, we identify its universal characteristic timescale levering o…
We derive an analytical pair potential of mean force for Brownian molecules in the liquid-state. Our approach accounts for many-particle correlations of crowding particles of the liquid, and for diffusive transport across the spatially modulated local density of crowders in the dense environment. Specializing on the limit of equal-size particles, we show that this diffusive transport leads to additional density- and structure-dependent terms in the interaction potential, and to a much stronger attraction (by a factor ~4 at average volume fraction of crowders 0.25) than in the standard depletio…
The Rayleigh model of nonlinear Brownian motion is revisited in which the heavy particle of mass M interacts with ideal gas molecules of mass m via instantaneous collisions. Using the van Kampen method of expansion of the master equation, non-linear corrections to the Fokker-Planck equation are obtained up to sixth order in the small parameter m/M, improving earlier results. The role and origin of non-Gaussian statistics of the random force in the corresponding Langevin equation are also discussed.
Collections of self-propelled particles that move persistently by continuously consuming free energy are a paradigmatic example of active matter. In these systems, unlike Brownian "hot colloids", the breakdown of detailed balance yields a continuous production of entropy at steady state, even for an ideal active gas. We quantify the irreversibility for a non-interacting active particle in two dimensions by treating both conjugated and time-reversed dynamics. By starting with underdamped dynamics, we identify a hidden rate of entropy production required to maintain persistence and prevent the r…
The Dynamic Density Functional (DDF) theory and standard Brownian dynamics simulations (BDS) are used to study the drifting effects of a colloidal particle in a polymer solution, both for ideal and interacting polymers. The structure of the stationary density distributions and the total induced current are analyzed for different drifting rates. We find good agreement with the BDS, which gives support to the assumptions of the DDF theory. The qualitative aspect of the density distribution are discussed and compared to recent results for driven colloids in one-dimensional channels and to analyti…
We report on the translation and rotation of particle clusters made through the combination of spherical building blocks. These clusters present ideal model systems to study the motion of objects with complex shape. Because they could be separated into fractions of well-defined configurations on a sufficient scale and their overall dimensions were below 300 nm, the translational and rotational diffusion coefficients of particle duplets, triplets and tetrahedrons could be determined by a combination of polarized dynamic light scattering (DLS) and depolarized dynamic light scattering (DDLS). The…
A small tagged particle immersed in a fluid exhibits the Brownian motion and diffuses at the long-time scale. Meanwhile, at the short-time scale, the dynamics of the tagged particle cannot be simply described by the usual generalized Langevin equation with the Gaussian noise, since the number of collisions between the tagged particle and fluid particles is rather small. At such a time scale, we should explicitly consider individual collision events between the tagged particle and the surrounding fluid particles. In this study, we analyzed the short-time dynamics of the tagged particle in an id…
The run-and-tumble dynamics of bacteria, as exhibited by \textit{E. coli}, offers a simple experimental realization of non-Brownian, yet diffusive, particles. Here we present some analytic and numerical results for models of the ideal (low-density) limit in which the particles have no hydrodynamic or other interactions and hence undergo independent motions. We address three cases: sedimentation under gravity; confinement by a harmonic external potential; and rectification by a strip of `funnel gates' which we model by a zone in which tumble rate depends on swim direction. We compare our result…
Solutions to the BBGKY hierarchy of equations for molecular Brownian particle in ideal gas are considered, and exact relations are derived between probability distribution of path of the particle, its derivatives in respect to gas density and irreducible many-particle correlations of gas atoms with the path. It is shown that all the correlations always give equally important contributions to evolution of the path distribution, and therefore the exact statistical mechanics theory does not reduce to classical kinetics even in the low-density limit.
We carry out a mathematically rigorous investigation into the equilibrium thermodynamics of massless and massive bosons confined in generalized Sierpinski carpets (GSCs), a class of infinitely ramified fractals having non-integer Hausdorff dimensions $d_h$. Due to the anomalous walk dimension $d_w>2$ associated with Brownian motion on GSCs, all extensive thermodynamic quantities are shown to scale with the spectral volume with dimension $d_s = 2(d_h/d_w)$ rather than the Hausdorff volume. We prove that for a low-temperature, high-density ideal massive Bose gas in an unbounded GSC, Bose-Einstei…
To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, a stochastic collision model is investigated. We consider the dynamics of a tracer particle of mass $M$, undergoing elastic collisions with ideal gas particles of mass $m$, in the Rayleigh limit $m<<M$. The probability density function (PDF) of the gas particle velocity is $f(\tilde{v}_m)$. Assuming a uniform collision rate and molecular chaos, we obtain the equilibrium distribution for the velocity of the tracer particle $W_{eq}(V_M)$. Depending on asymptotic properties of $f(\tilde{v}_m)$ we fi…
We review the quantum version of the linear Boltzmann equation, which describes in a non-perturbative fashion, by means of scattering theory, how the quantum motion of a single test particle is affected by collisions with an ideal background gas. A heuristic derivation of this Lindblad master equation is presented, based on the requirement of translation-covariance and on the relation to the classical linear Boltzmann equation. After analyzing its general symmetry properties and the associated relaxation dynamics, we discuss a quantum Monte Carlo method for its numerical solution. We then revi…
