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.
Levy solutions of a randomly forced Burgers equation
考虑一维Burgers方程在空间上受布朗运动和时间上受白噪声过程$\partial_t u + u \partial_x u = f(x,t)$的强迫，利用$2E(f(x,t)f(y,s)) = (|x|+|y|-|x-y|)δ(t-s)$证明了存在Levy过程解，并给出了Levy过程解的特征指数的演化方程.特别地，我们给出了$u_0(x)=0$情形的显式解.
The problem of one-dimensional randomly forced Burgers turbulence is considered in terms of (1+1) directed polymers. In the limit of strong turbulence (which corresponds to the zero temperature limit for the directed polymer system) using the replica technique a general explicit expression for the joint distribution function of two velocities separated by a finite distance is derived. In particular, it is shown that at length scales much smaller than the injection length of the Burgers random force the moments of the velocity increment exhibit typical strong intermittency behavior.
Evaluation of the rate constant and deposition velocity for the escape of Brownian particles over potential barriers
我们以类似于Komrasekhar的方式使用Fokker-Planck方程分析布朗粒子越过势垒的逃逸（Rev. Modern Phys.，1943），推导出作为颗粒响应时间的函数的颗粒到表面的沉积速度的公式。对于非常小的颗粒响应时间，颗粒沉积速度降低到使用Smoluchowski方程的准稳态解获得的速度，并且对于非常大的颗粒响应时间，沉积速度对应于基于过渡态方法（E.魏格纳，法拉第社，1938年）。
We study $N$ vicious Brownian bridges propagating from an initial configuration $\{a_1 < a_2 < \ldots< a_N \}$ at time $t=0$ to a final configuration $\{b_1 < b_2 < \ldots< b_N \}$ at time $t=t_f$, while staying non-intersecting for all $0\leq t \leq t_f$. We first show that this problem can be mapped to a non-intersecting Dyson's Brownian bridges with Dyson index $β=2$. For the latter we derive an exact effective Langevin equation that allows to generate very efficiently the vicious bridge configurations. In particular, for the flat-to-flat configuration in the large $N$ limit, where $a_i = b…
We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equati…
Symmetries constrain dynamics. We test this fundamental physical principle, experimentally and by molecular dynamics simulations, for a hot Janus swimmer operating far from thermal equilibrium. Our results establish scalar and vectorial steady-state fluctuation theorems and a thermodynamic uncertainty relation that link the fluctuating particle current to its entropy production at an effective temperature. A Markovian minimal model elucidates the underlying non-equilbrium physics.
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…
We present a numerical scheme for simulating the dynamics of Brownian particles suspended in a fluid. The motion of the particles is tracked by the Langevin equation, whereas the host fluid flow is analyzed by using the lattice Boltzmann method. The friction force between a particle and the fluid is evaluated correctly based on the velocity difference at the position of the particle. The coupling method accurately reproduces the long-time tail observed in the velocity auto-correlation function. We also show that the fluctuation-dissipation relation holds between the relaxation of a single part…
Motivated by the simple models of molecular motor obeying a linear force-velocity relation, we have studied the stochastic dynamics of a Brownian particle in the presence of a linear velocity dependent force, $f_s(1-\frac{v}{v_0})$ where $f_{s}$ is a constant. The position and velocity auto correlation functions in different situations of the dynamics are calculated exactly. We observed that the velocity auto correlation function shows an exponentially decaying behaviour with time and saturates to a constant value in the time asymptotic limit, for a fixed $f_s$. It attains saturation faster wi…
We study the Brownian dynamics of a solid particle on a vibrating solid surface. Phenomenologically, the interaction between the two solid surfaces is modeled by solid friction, and the Gaussian white noise models the vibration of the solid surface. The solid friction is proportional to the sign of relative velocity. We derive the Fokker-Planck (FP) equation for the time-dependent probability distribution to find the particle at a given location. We calculate analytically the steady state velocity distribution function, mean-squared velocity and diffusion coefficient in $d$ dimensions. We pres…
Recently, it has been discovered that systems of Active Brownian particles (APB) at high density organise their velocities into coherent domains showing large spatial structures in the velocity field. Such a collective behavior occurs spontaneously, i.e. is not caused by any specific interparticle force favoring the alignment of the velocities. This phenomenon was investigated in the absence of thermal noise and in the overdamped regime where inertial forces could be neglected. In this work, we demonstrate through numerical simulations and theoretical analysis that the velocity alignment is a …
We obtain an exact result for the midpoint probability distribution function (pdf) of the stationary continuum directed polymer, when averaged over the disorder. It is obtained by relating that pdf to the linear response of the stochastic Burgers field to some perturbation. From the symmetries of the stochastic Burgers equation we derive a fluctuation-dissipation relation so that the pdf gets given by the stationary two space-time points correlation function of the Burgers field. An analytical expression for the latter was obtained by Imamura and Sasamoto [2013], thereby rendering our result e…
Multistability is one of the most important phenomena in dynamical systems, e.g. bistability enables the implementation of logic gates and therefore computation. Recently multistability has attracted a greatly renewed interest related to memristors and graphene structures, to name only a few. We investigate tristability in velocity dynamics of a Brownian particle subjected to a tilted periodic potential. It is demonstrated that the origin of this effect is attributed to the arcsine law for the velocity dynamics at the zero temperature limit. We analyze the impact of thermal fluctuations and co…
