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We study in phase space a zero-dimensional system of Brownian particles which move in a periodic potential and subject to an internal time derivative Ornstein–Uhlenbeck noise. To resolve the Fokker–Planck equation in such a case, we propose an approximate analytical method. The theoretical predictions exhibit a second order noise-induced nonequilibrium phase transition, which is confirmed by numerical simulation results. The phase transition brings the system from an ergodicity to a nonergodicity phase as the potential barrier height decreases.

We investigate the low-temperature statistical properties of a harmonic oscillator coupled to a heat bath, where the low-frequency spectrum vanishes. We obtain the exact result of the zero point energy. Due to the low frequency shortage of environmental oscillators' spectral density, the coordinate and momentum correlation functions decay as τ−4 and τ−6 respectively at zero temperature, where τ is the correlation time. The low-temperature behavior of the mean energy does not violate the third law of thermodynamics, but differs largely from the Ohmic spectrum case.

We propose an n-order noise, which is realized by driving an n-order linear differential equation with a Gaussian white noise. The time-derivative noise is a low-resistant band-passing noise. If the derivative noise is regarded as a thermal one, the system has a vanishing effective friction and it should induce ballistic diffusion of a free particle at long times. The simulation method for the generalized Langevin equation driven by the n-order noise is discussed systematically. The features of three-order derivative noises are presented when they are applied to a ratchet system.

The generalized Langevin equation is used to describe the motion of a charged particle interacting with a blackbody radiation field via dipole coupling. The exact expressions for the mean-square displacement and velocity of such a particle are obtained, which show a ballistic diffusion and a modified Kubo fluctuation–dissipation relation. In particular, a velocity-dependent coupling or force can induce this novel phenomenon.

For a harmonic oscillator coupled to a heat bath with harmonic velocity noise and harmonic noise, we obtain the exact average energy by resoving the corresponding generalized quantum Langevin equation. Due to the different power spectra of the two kinds of noise, the harmonic oscillator driven by external harmonic force exhibits different resonant features. This is an approach for examing the characterization of internal noise.

Based on the Faddeev formalism of path-integral quantization for a constrained Hamiltonian system, the Casimir effect between two non-parallel lines in the (2+1)-dimensional space is calculated by using conformal mapping and Plana summation formula in the theory of complex variable function. Without introducing any cutoff of parameter, the finite analytical expression is obtained.