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Register a new userDynamics of the Desai-Zwanzig model in multi-well and random energy landscapes
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We analyze a variant of the Desai-Zwanzig model [J. Stat. Phys. {bf 19}1-24 (1978)]. In particular, we study stationary states of the mean field limit for a system of weakly interacting diffusions moving in a multi-well potential energy landscape, coupled via a Curie-Weiss type (quadratic) interaction potential. The location and depth of the local minima of the potential are either deterministic or random. We characterize the structure and nature of bifurcations and phase transitions for this system, by means of extensive numerical simulations and of analytical calculations for an explicitly solvable model. Our numerical experiments are based on Monte Carlo simulations, the numerical solution of the time-dependent nonlinear Fokker-Planck (McKean-Vlasov equation), the minimization of the free energy functional and a continuation algorithm for the stationary solutions.
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We introduce a diffusion model for energetically inhomogeneous systems. A random walker moves on a spin-S Ising configuration, which generates the energy landscape on the lattice through the nearest-neighbors interaction. The underlying energetic environment is also made dynamic by properly coupling the walker with the spin lattice. In fact, while the walker hops across nearest-neighbor sites, it can flip the pertaining spins, realizing a diffusive dynamics for the Ising system. As a result, the walk is biased towards high energy regions, namely the boundaries between clusters. Besides, the coupling introduced involves, with respect the ordinary diffusion laws, interesting corrections depending on either the temperature and the spin magnitude. In particular, they provide a further signature of the phase-transition occurring on the magnetic lattice.
In this paper we first analyzed the inductive bias underlying the data scattered across complex free energy landscapes (FEL), and exploited it to train deep neural networks which yield reduced and clustered representation for the FEL. Our parametric method, called Information Distilling of Metastability (IDM), is end-to-end differentiable thus scalable to ultra-large dataset. IDM is also a clustering algorithm and is able to cluster the samples in the meantime of reducing the dimensions. Besides, as an unsupervised learning method, IDM differs from many existing dimensionality reduction and clustering methods in that it neither requires a cherry-picked distance metric nor the ground-true number of clusters, and that it can be used to unroll and zoom-in the hierarchical FEL with respect to different timescales. Through multiple experiments, we show that IDM can achieve physically meaningful representations which partition the FEL into well-defined metastable states hence are amenable for downstream tasks such as mechanism analysis and kinetic modeling.
We investigate slow-roll inflation in a multi-field random Gaussian landscape. The landscape is assumed to be small-field, with a correlation length much smaller than the Planck scale. Inflation then typically occurs in small patches of the landscape, localized near inflection or saddle points. We find that the inflationary track is typically close to a straight line in the field space, and the statistical properties of inflation are similar to those in a one-dimensional landscape. This picture of multi-field inflation is rather different from that suggested by the Dyson Brownian motion model; we discuss the reasons for this difference. We also discuss tunneling from inflating false vacua to the neighborhood of inflection and saddle points and show that the tunneling endpoints tend to concentrate along the flat direction in the landscape.
One of the most challenging and frequently arising problems in many areas of science is to find solutions of a system of multivariate nonlinear equations. There are several numerical methods that can find many (or all if the system is small enough) solutions but they each exhibit characteristic problems. Moreover, traditional methods can break down if the system contains singular solutions. Here, we propose an efficient implementation of Newton homotopies, which can sample a large number of the stationary points of complicated many-body potentials. We demonstrate how the procedure works by applying it to the nearest-neighbor $phi^4$ model and atomic clusters.
The dynamics of the one-dimensional random transverse Ising model with both nearest-neighbor (NN) and next-nearest-neighbor (NNN) interactions is studied in the high-temperature limit by the method of recurrence relations. Both the time-dependent transverse correlation function and the corresponding spectral density are calculated for two typical disordered states. We find that for the bimodal disorder the dynamics of the system undergoes a crossover from a collective-mode behavior to a central-peak one and for the Gaussian disorder the dynamics is complex. For both cases, it is found that the central-peak behavior becomes more obvious and the collective-mode behavior becomes weaker as $K_{i}$ increase, especially when $K_{i}>J_{i}/2$ ($J_{i}$ and $K_{i}$ are exchange couplings of the NN and NNN interactions, respectively). However, the effects are small when the NNN interactions are weak ($K_{i}<J_{i}/2$).
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