Equilibrium Fluctuations for the Totally Asymmetric Zero-Range Process

We consider the one-dimensional Totally Asymmetric Zero-Range process evolving on ? and starting from the Geometric product measure ν _ρ . On the hyperbolic time scale the temporal evolution of the limit density fluctuation field is deterministic, in the sense that the limit field at time t is a translation of the initial one. We consider the system in a reference frame moving at this velocity and we show that the limit density fluctuation field does not evolve in time until N ^4/3, which implies the current across a characteristic to vanish on this longer time scale.     

Renewal-anomalous-heterogeneous files

Renewal-anomalous-heterogeneous files are solved. A simple file is made of Brownian hard spheres that diffuse stochastically in an effective 1D channel. Generally, Brownian files are heterogeneous: the spheres' diffusion coefficients are distributed and the initial spheres' density is non-uniform. In renewal-anomalous files, the distribution of waiting times for individual jumps is not exponential as in Brownian files, yet obeys: ψα(t)∼t−1−α, 0<α<1. The file is renewal as all the particles attempt jumping at the same time. It is shown that the mean square displacement (MSD) in a renewal-anomalous-heterogeneous file, 〈r²〉, obeys, , where nrml〈r²〉 is the MSD in the corresponding Brownian file. This scaling is an outcome of an exact relation (derived here) connecting probability density functions of Brownian files and renewal-anomalous files. It is also shown that non-renewal-anomalous files are slower than the corresponding renewal ones.     

Statistical Properties of the Burgers Equation with Brownian Initial Velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its n-point correlations. In the same limit, we derive the n-point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.     

Fluctuations in the Thermodynamic Limit of Focussing Cubic Schrödinger

We consider the statistical mechanics of a complex field Z whose dynamics are governed by the focussing cubic Schrödinger equation. Here the Hamiltonian $$H = \int {_\Omega } \left[ {\frac{1}{2}\left| {\nabla Z} \right|^2 - \frac{1}{4}\left| Z \right|^4 } \right]dx$$ is unbounded from below, preventing the natural Gibbs measure from being normalizable. This difficulty may be circumvented⁽⁵⁾ by taking Ω the circle of perimeter L and fixing the mean-square (which is conserved by the dynamics): $\int {_0^L } \left| Z \right|^2 dx = LD$ for positive “density” D. The resulting (probability) measure on paths is absolutely continuous to the two-dimensional Wiener measure and is known to be invariant under the flow.^{(2, 7)} One way to extend this picture to the whole-line flow is to take the thermodynamic limit (L↑∞). Unfortunately, the unboundedness of H causes vast local concentration of the field as L increases and leads to collapse at L=∞.⁽¹¹⁾ Here we attempt to capture fluctuations away from this collapse by performing a joint continuum and infinite-volume limit for an appropriate lattice ensemble. The result is that, for high density, the scaled paths go over into a White Noise.     

Continuous sample paths in quantum field theory

Nelson's free Markoff field on ? ^l+1 is a natural generalization of the Ornstein-Uhlenbeck process on ?¹, mapping a class of distributions φ(x,t) on ? ^l ×?¹ to mean zero Gaussian random variables φ with covariance given by the inner product \(\left( {\left( {m^2 - \Delta - \frac{{\partial ^2 }}{{\partial t^2 }}} \right)^{ - 1} \cdot , \cdot } \right)_2 \) . The random variables φ can be considered functions φ〈q〉=∝ φ(x,t)q(x,t)d x dt on a space of functionsq(x,t). In the O.U. case,l=0, the classical Wiener theorem asserts that the underlying measure space can be taken as the space of continuous pathst →q(t). We find analogues of this, in the casesl>0, which assert that the underlying measure space of the random variables φ which have support in a bounded region of ? ^l+1 can be taken as a space of continuous pathst →q(·,t) taking values in certain Soboleff spaces.     

Hamiltonian and Brownian systems with long-range interactions: V. Stochastic kinetic equations and theory of fluctuations

We developed a theory of fluctuations for Brownian systems with weak long-range interactions. For these systems, there exists a critical point separating a homogeneous phase from an inhomogeneous phase. Starting from the stochastic Smoluchowski equation governing the evolution of the fluctuating density field of Brownian particles, we determine the expression of the correlation function of the density fluctuations around a spatially homogeneous equilibrium distribution. In the stable regime, we find that the temporal correlation function of the Fourier components of density fluctuations decays exponentially rapidly, with the same rate as the one characterizing the damping of a perturbation governed by the deterministic mean field Smoluchowski equation (without noise). On the other hand, the amplitude of the spatial correlation function in Fourier space diverges at the critical point T=T_c (or at the instability threshold k=k_m) implying that the mean field approximation breaks down close to the critical point, and that the phase transition from the homogeneous phase to the inhomogeneous phase occurs sooner. By contrast, the correlations of the velocity fluctuations remain finite at the critical point (or at the instability threshold). We give explicit examples for the Brownian Mean Field (BMF) model and for Brownian particles interacting via the gravitational potential and via the attractive Yukawa potential. We also introduce a stochastic model of chemotaxis for bacterial populations generalizing the deterministic mean field Keller-Segel model by taking into account fluctuations and memory effects.     

Brownian motion in a fluid in elongational flow

Brownian motion of a spherical particle in stationary elongational flow is studied. We derive the Langevin equation together with the fluctuation-dissipation theorem for the particle from nonequilibrium fluctuating hydrodynamics to linear order in the elongation-rate-dependent inverse penetration depths. We then analyze how the velocity autocorrelation function as well as the mean square displacement are modified by the elongational flow. We find that for times small compared to the inverse elongation rate the behavior is similar to that found in the absence of the elongational flow. Upon approaching times comparable to the inverse elongation rate the behavior changes and one passes into a time domain where it becomes fundamentally different. In particular, we discuss the modification of thet ^–3/2 long-time tail of the velocity autocorrelation function and comment on the resulting contribution to the mean square displacement. The possibility of defining a diffusion coefficient in both time domains is discussed.     

Diffusion in a bistable potential. General inverse friction expansion

The derivation of the characteristic times and of the density probability distribution for the motion of a Brownian particle in a bistable potential at intermediate friction was, until now, essentially limited to low orders in the inverse frictionγ ^?1. On the other hand, at least for temperatures low with respect to the barrier height, the Kramers time, which is the lowest nonzero eigenvalue in the bistable potential problem, is known exactly. This paper presents a systematic approach for the determination of the solution of the Fokker-Planck equation in an arbitrary potential in the overdamped regime. This calculation includes anharmonicity corrections up to orderγ ^?5. One feature of this paper is to show that the problem is equivalent to replacing the original potentialφ(x) by a free energy which, for a velocity distribution at equilibrium, simply is \(\widetilde\phi \) =φ(x) ?k _BT ln[g(x)], where $$g(x) = \left\{ {{{m\gamma } \mathord{\left/ {\vphantom {{m\gamma } {[2\phi ''(x)]}}} \right. \kern-\nulldelimiterspace} {[2\phi ''(x)]}}} \right\}\left\{ {1 - [{{1 - 4\phi ''(x)} \mathord{\left/ {\vphantom {{1 - 4\phi ''(x)} {m\gamma ^2 }}} \right. \kern-\nulldelimiterspace} {m\gamma ^2 }}]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } \right\}$$ For out-of-equilibrium velocity distribution an effective potential is also explicitly given. In every case the function g(x) plays a crucial role. This approach is then applied to the exact determination, in the low-temperature limit, of all the characteristic times and of the probability distribution in bistable potentials. Moreover, from the knowledge of the characteristic times and probability density distribution, it would be easy to determine the general and exact Suzuki scaling law for the relaxation from the instability point at intermediate friction.     

Dynamical correlations for vicious random walk with a wall

A one-dimensional system of nonintersecting Brownian particles is constructed as the diffusion scaling limit of Fisher's vicious random walk model. N Brownian particles start from the origin at time t=0 and undergo mutually avoiding motion until a finite time t=T. Dynamical correlation functions among the walkers are exactly evaluated in the case with a wall at the origin. Taking an asymptotic limit N→∞, we observe discontinuous transitions in the dynamical correlations. It is further shown that the vicious walk model with a wall is equivalent to a parametric random matrix model describing the crossover between the Bogoliubov–deGennes universality classes.     

Stochastic processes originating in deterministic microscopic dynamics

We investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics, i.e., ifQ(t) is the displacement of the test particle we letQ _A(t) =Q(At)/√A and consider the distribution of the trajectory Q_A(t) in the limit A→∞. The randomness of the motion is due entirely to the randomness of the initial state of the fluid, test particle, or both, and the process is generally non-Markovian. Nevertheless, it can be proven in some cases and we expect it to be true in many more that Q_A (t) looks like Brownian motion in the limit A→∞. Some results for simple model systems are presented.     

The nonlinear Schrödinger limit of the Zakharov equations governing Langmuir turbulence

We consider the initial value problem for the Zakharov equations $$\begin{gathered} \left( Z \right)\frac{1}{{\lambda ^2 }}n_{tt} - \Delta (n + \left| {\rm E} \right|^2 ) = 0n(x,0) = n_0 (x) \hfill \\ n_t (x,0) = n_1 (x) \hfill \\ iE_t + \Delta E - nE = 0E(x,0) = E_0 (x) \hfill \\ \end{gathered} $$ (x∈? ^k ,k=2, 3,t ≧0) which model the propagation of Langmuir waves in plasmas. For suitable initial data solutions are shown to exist for a time interval independent of λ, a parameter proportional to the ion acoustic speed. For such data, solutions of (Z) converge as λ → ∞ to a solution of the cubic nonlinear Schrödinger equation (CSE)iE _t +ΔE+|E|² E=0. We consider both weak and strong solutions. For the case of strong solutions the results are analogous to previous results on the incompressible limit of compressible fluids.     

Single-vacancy induced motion of a tracer particle in a two-dimensional lattice gas

We study the random motion of a tracer particle in a two-dimensional dense lattice gas. Repeated encounters of asingle vacancy displace the tracer particle from its initial position by a vector y of which we calculate the time-dependent distributionP _t(y). On an infinite lattice and for large times $$P_t (y) \simeq \frac{{2(\pi - 1)}}{{\ln t}}K_0 \left( {\left( {\frac{{4\pi (\pi - 1)}}{{\ln t}}} \right)^{1/2} y} \right)$$ whereK ₀ is a modified Bessel function. The same problem is studied on a finiteL×L lattice with periodic boundary conditions; thereP _t(y) is shown to be a Gaussian on a time scaleL ² InL. On an ∞×L strip and for large times,P _t(y) is an explicitly given (but nonelementary) function of the scaling variable ξy ₁/t ^1/4, identical to the function occurring in the problem of a random walker on a random one-dimensional path.     

the distribution of exit times for weakly colored noise

We analyze the exit time (first passage time) problem for the Ornstein-Uhlenbeck model of Brownian motion. Specifically, consider the positionX(t) of a particle whose velocity is an Ornstein-Uhlenbeck process with amplitudeσ/ρ and correlation time ε², $$dX/dt = \sigma Z/\varepsilon , dZ/dt = - Z/\varepsilon ^2 + 2^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \xi (t)/\varepsilon $$ whereξ(t) is Gaussian white noise. Let the exit timet _ex be the first time the particle escapes an interval ?A, given that it starts atX(0)=0 withZ(0)=z ₀. Here we determine the exit time probability distributionF(t)≡Prob {t _ex>t} by directly solving the Fokker-Planck equation. In brief, after taking a Laplace transform, we use singular perturbation methods to reduce the Fokker-Planck equation to a boundary layer problem. This boundary layer problem turns out to be a half-range expansion problem, which we solve via complex variable techniques. This yields the Laplace transform ofF(t) to within a transcendentally smallO(e ^?A/εσ +e ^?B/εσ error. We then obtainF(t) by inverting the transform order by order in ε. In particular, by lettingB→∞ we obtain the solution to Wang and Uhlenbeck's unsolved problem b; throughO(ε²σ²/A ¹) this solution is $$F(t) = Erf\left\{ {\frac{{A + \varepsilon \sigma \alpha + \varepsilon \sigma z_0 }}{{2\sigma (t - \varepsilon ^2 \kappa )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} }}} \right\} + ... for \frac{t}{{\varepsilon ^2 }} > > 1$$ andF=1 otherwise. Here, α=∥ξ(1/2)∥=1.4603?, where ξ is the Riemann zeta function, and the constant κ is 0.22749?.     

Phase transition for absorbed Brownian motion with drift

We study one-dimensional Brownian motion with constant drift toward the origin and initial distribution concentrated in the strictly positive real line. We say that at the first time the process hits the origin, it is absorbed. We study the asymptotic behavior, ast, ofm _t , the conditional distribution at time zero of the process conditioned on survival up to timet and on the process having a fixed value at timet. We find that there is a phase transition in the decay rate of the initial condition. For fast decay rate (subcritical case)m _t is localized, in the critical casem _t is located around , and for slow rates (supercritical case)m _t is located aroundt. The critical rate is given by the decay of the minimal quasistationary distribution of this process. We also study in each case the asymptotic distribution of the process, scaled by , conditioned as before. We prove that in the subcritical case this distribution is a Brownian excursion. In the critical case it is a Brownian bridge attaining 0 for the first time at time 1, with some initial distribution. In the supercritical case, after centering around the expected value—which is of the order oft—we show that this process converges to a Brownian bridge arriving at 0 at time 1 and with a Gaussian initial distribution.     

Kinetics of particle coarsening processes

A nonequilibrium statistical mechanical theory for particle coarsening processes is presented. In this theory, the rate of change of a given particle is determined by both a deterministic and a fluctuation terms, and the particle size distribution (PSD) satisfies Fokker-Planck-type equation. We use a time scaling technique and find the PSD scaled by average particle size as well as the power laws of time dependence of some quantities. The asymptotic scaled PSD is independent of initial condition but does depend on the equilibrium volume fraction. We show that the average radius grows att ^1/3 and the density of particles decays ast ^?1.     

Cut contributions in the triple-pomeron limit

We investigate pomeron cut contributions in the triple-pomeron limit of one-particle inclusive cross sections, near t = 0 where the triple-pole coupling vanishes. We find that at t = 0 the cuts themselves are suppressed, contributing factors (lnM²)^?2, rather than the single logarithms characteristic of cut contributions in two-body processes. We construct a simple reggeon calculus model in which all important cuts near t = 0 can be calculated, and suggest a simple way of parametrizing the data in that region.     

Evolution Law of the Negative Binomial State in Laser Channel and its Photon-Number Decay Formula

For the first time we examine how a negative binomial state (NBS), whose density operator is \({\sum }_{n=0}^{\infty }\frac {\left (n+s\right ) !} {n!s!}\gamma ^{s+1}\left (1-\gamma \right )^{n}\left \vert n\right \rangle \left \langle n\right \vert ,\) evolves in a laser channel. By using a newly derived generating function formula about Laguerre polynomial we obtain the evolution law of NBS, which turns out to be an infinite operator-sum of photon-added negative binomial state with a new negative-binomial parameter, and the photon number of NBS decays with e ^?2(κ?g)t , where g and κ represent the cavity gain and loss respectively. The technique of integration (summation) within an ordered product of operators is used in our discussions.     

Ballistic diffusion induced by non-Gaussian noise

In this letter,we have analyzed the diffusive behavior of a Brownian particle subject to both internal Gaussian thermal and external non-Gaussian noise sources.We discuss two time correlation functions C(t) of the non-Gaussian stochastic process,and find that they depend on the parameter q,indicating the departure of the non-Gaussian noise from Gaussian behavior:for q ≤ 1,C(t) is fitted very well by the first-order exponentially decaying curve and approaches zero in the longtime limit,whereas for q 1,C(t) can be approximated by a second-order exponentially decaying function and converges to a non-zero constant.Due to the properties of C(t),the particle exhibits a normal diffusion for q ≤ 1,while for q 1 the non-Gaussian noise induces a ballistic diffusion,i.e.,the long-time mean square displacement of the free particle reads [x(t)-x(t)]2∝t2.     

Rigorous treatment of metastable states in the van der Waals-Maxwell theory

We consider a classical system, in a ν-dimensional cube Ω, with pair potential of the formq(r) + γ ^v φ(γr). Dividing Ω into a network of cells ω₁, ω₂,..., we regard the system as in a metastable state if the mean density of particles in each cell lies in a suitable neighborhood of the overall mean densityρ, withρ and the temperature satisfying $$f_0 (\rho ) + \tfrac{1}{2}\alpha \rho ^2 > f(\rho ,0 + )$$ and $$f''_0 (\rho ) + 2\alpha > 0$$ wheref(ρ, 0+) is the Helmholz free energy density (HFED) in the limit γ→ 0; α = ∫ φ(r)d ^v r andf ₀ (ρ) is the HFED for the caseφ = 0. It is shown rigorously that, for periodic boundary conditions, the conditional probability for a system in the grand canonical ensemble to violate the constraints at timet > 0, given that it satisfied them at time 0, is at mostλt, whereλ is a quantity going to 0 in the limit $$|\Omega | \gg \gamma ^{ - v} \gg |\omega | \gg r_0 \ln |\Omega |$$ Here,r ₀ is a length characterizing the potentialq, andx ? y meansx/y → +∞. For rigid walls, the same result is proved under somewhat more restrictive conditions. It is argued that a system started in the metastable state will behave (over times ?λ ^?1) like a uniform thermodynamic phase with HFED f₀(ρ) + 1/2αρ², but that having once left this metastable state, the system is unlikely to return.