The aim of this mini-review article is to clarify the relation between two distinct formulations of the thermodynamic free energy for collective variables which can be found in the molecular dynamics literature. In doing so, we discuss the different ensemble concepts underlying the two definitions and reveal their relation to strong confinement (restraints) and molecular constraints. The latter analysis is based on a variant of Federer's coarea formula which can be regarded as a generalization of Fubini's theorem for iterated integrals to curvilinear coordinates and which implies the famous "blue moon" ensemble identity for computing conditional expectations using constrained simulations. For illustration we will present a few paradigmatic examples.
The chemical master equation (CME) is the fundamental evolution equation of the stochastic description of biochemical reaction kinetics. In most applications it is impossible to solve the CME directly due to its high dimensionality. Instead indirect approaches based on realizations of the underlying Markov jump process are used such as the stochastic simulation algorithm (SSA). In the SSA, however, every reaction event has to be resolved explicitly such that it becomes numerically inefficient when the system's dynamics include fast reaction processes or species with high population levels. In many hybrid approaches, such fast reactions are approximated as continuous processes or replaced by quasi-stationary distributions either in a stochastic or deterministic context. Current hybrid approaches, however, almost exclusively rely on the computation of ensembles of stochastic realizations. We present a novel hybrid stochastic-deterministic approach to solve the CME directly. Starting point is a partitioning of the molecular species into discrete and continuous species that induces a partitioning of the reactions into discrete-stochastic and continuous-deterministic. The approach is based on a WKB approximation of a conditional probability distribution function (PDF) of the continuous species (given a discrete state) combined with a multiscale expansion of the CME. The black resulting hybrid stochastic-deterministic evolution equations comprise a CME with averaged propensities for the PDF of the discrete species that is coupled to an evolution equation of the partial expectation of the continuous species for each discrete state. In contrast to indirect hybrid methods, the impact of the evolution of discrete species on the dynamics of the continuous species has to be taken into account explicitly. The proposed approach is efficient whenever the number of discrete molecular species is small. We illustrate the performance of the new hybrid stochastic-deterministic approach in application to model systems of biological interest.
In this paper we revisit the problem of Brownian motion in a tilted periodic potential. We use homogenization theory to derive general formulas for the effective velocity and the effective diffusion tensor that are valid for arbitrary tilts. Furthermore, we obtain power series expansions for the velocity and the diffusion coefficient as functions of the external forcing. Thus, we provide systematic corrections to Einstein's formula and to linear response theory. Our theoretical results are supported by extensive numerical simulations. For our numerical experiments we use a novel spectral numerical method that leads to a very efficient and accurate calculation of the effective velocity and the effective diffusion tensor.
We study optimal control of diffusions with slow and fast variables and address a question raised by practitioners: is it possible to first eliminate the fast variables before solving the optimal control problem and then use the optimal control computed from the reduced-order model to control the original, high-dimensional system? The strategy "first reduce, then optimize"-rather than "first optimize, then reduce"-is motivated by the fact that solving optimal control problems for high-dimensional multiscale systems is numerically challenging and often computationally prohibitive. We state sufficient and necessary conditions, under which the "first reduce, then control" strategy can be employed and discuss when it should be avoided. We further give numerical examples that illustrate the "first reduce, then optmize" approach and discuss possible pitfalls.
The effect of weak lateral dispersion of Zakharov-Kutznetsov-type on a Benjamin-Ono solitary wave is studied both asymptotically and numerically. The asymptotic solution is based on an approximate variational solution for the solitary wave, which is then modulated in time through the use of conservation equations. The effect of the dispersive radiation shed as the solitary wave evolves is also included in the modulation equations. It is found that the weak lateral dispersion produces a strongly anisotropic, stable solitary wave which decays algebraically in the direction of propagation, as for the Benjamin-Ono solitary wave, and exponentially in the transverse direction. Moreover, it is found that initial conditions with amplitude above a threshold evolve into solitary waves, while those with amplitude below the threshold evolve as lumps for a short time, then merge into radiation. The modulation equations are found to give a quantitatively accurate description of the evolution of an initial condition into an anisotropic solitary wave. The existence of stable solitary waves is in contrast to previous studies of Benjamin-Ono-type equations subject to the stronger Kadomstev-Petviashvili or Benjamin-Ono-type lateral dispersion, for which the solitary waves either decay or collapse. The present study then completes the catalog of possible behaviors under lateral dispersion.
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