where z denotes the microscopic configuration and ρeq(z)dz is the equilibrium measure for the microscopic dynamics. In some cases the actual Hamiltonian (total energy) may be expressible in terms of the coarse-grained variables and is strictly conserved, but this is not always the case since some of the energy may in fact be incorporated in the coarse-grained free energy as an “entropy”. The eq...

We consider a class of Langevin diffusions with state-dependent volatility. The volatility of the diffusion is chosen so as to make the stationary distribution of the diffusion with respect to its natural clock, a heated version of the stationary density of interest. The motivation behind this construction is the desire to construct uniformly ergodic diffusions with required stationary densitie...

Langevin mode theory and the coarse-grained elastic network model (ENM) for proteins are combined to yield the Langevin network model (LNM). Hydrodynamic radii of 6 A were assigned to each alpha-carbon on the basis of matching experimental translational and rotational diffusion constants of lysozyme, myoglobin, and hemoglobin with those calculated using a rigid body bead model with hydrodynamic...

A framework recently developed for the extraction of a dynamic reaction coordinate to mediate reactions buried in a multidimensional Langevin equation is extended to the generalized Langevin equations without a priori assumption of the forms of the potential (in general, nonlinearly coupled systems) and the friction kernel. The equation of motion with memory effect can be transformed into an eq...

Above a critical temperature, a supported single domain ferromagnetic particle responds to an applied magnetic field as if it were a paramagnet with a very large spin. Its average magnetization is given by the Langevin equation as expected from simple thermodynamic considerations. The average magnetization of an ensemble of unsupported ferromagnetic clusters also approximately follows the Lange...

In this paper we derive exact quantum Langevin equations for stochastic dynamics of large-scale inflation in de Sitter space. These quantum Langevin equations are the equivalent of the Wigner equation and are described by a system of stochastic differential equations. We present a formula for the calculation of the expectation value of a quantum operator whose Weyl symbol is a function of the l...

When simulating molecular systems using deterministic equations of motion (e.g., Newtonian dynamics), such equations are generally numerically integrated according to a well-developed set of algorithms that share commonly agreed-upon desirable properties. However, for stochastic equations of motion (e.g., Langevin dynamics), there is still broad disagreement over which integration algorithms ar...

We study the connection between Hamiltonian dynamics and irreversible, stochastic equations, such as the Langevin equation. We consider a simple model of a classical harmonic oscillator (Brownian particle) coupled to a field (heat bath). In traditional approaches Langevin-type equations have been derived from Hamiltonian dynamics, but these equations contain memory terms and are both time-rever...

We study the detailed path-wise behavior of the discrete-time Langevin algorithm for nonconvex Empirical Risk Minimization (ERM) through the lens of metastability, adopting some techniques from Berglund and Gentz (2003). For a particular local optimum of the empirical risk, with an arbitrary initialization, we show that, with high probability, one of the two mutually exclusive events will occur...

Magnetic Particle Imaging (MPI) is a promising new tracer modality with zero attenuation in tissue, high contrast and sensitivity, and an excellent safety profile. However, the spatial resolution of MPI is currently around 1 mm in small animal scanners. Especially considering tradeoffs when scaling up MPI scanning systems to human size, this resolution needs to be improved for clinical applicat...