![]() ![]() If the external potential is conservative and the noise term derives from a reservoir in thermal equilibrium, then the long-time solution to the Langevin equation must reduce to the Boltzmann distribution, which is the probability distribution function for particles in thermal equilibrium. It was rstbrought to popular attention in 1827 by the Scottish botanist Robert Brown, who noticed that pollen grainssuspended in water move about at random even when the water appears to be very still. At long times, the velocity remains nonzero, and the position and velocity distributions correspond to that of thermal equilibrium. Brownian motion is perhaps the most important stochastic process we will see in this course. For nonzero temperatures, the velocity can be kicked to values higher than the initial value due to thermal fluctuations. ‘Brownian Motionby M¨orters and Peres, a modern and attractive account of one of the central topicsof probability theory, will serve both as an accessible introduction at the level of a Master’s courseand as a work of reference for ne properties of Brownian paths. At zero temperature, the velocity rapidly decays from its initial value (the red dot) to zero due to damping. The right panel captures the corresponding equilibrium probability distributions. ![]() The left panel shows the time evolution of the phase portrait of a harmonic oscillator at different temperatures. This plot corresponds to solutions of the complete Langevin equation obtained using the Euler–Maruyama method. If it is initially located at the origin with probability 1, then the result is ⟨ η i ( t ) η j ( t ′ ) ⟩ = 2 λ k B T δ i, j δ ( t − t ′ ) . A local limit theorem for convergence of probability density functions is provided as a tool for the computation of hitting time distributions for Brownian motion, with or without drift, as a limit of hitting times for random walk, and other asymptotic limit theorems of this nature.
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