Temporal disconnectivity of the energy landscape in glassy systems

Abstract

An alternative graphical representation of the potential energy landscape (PEL) has been developed and applied to a binary Lennard-Jones glassy system, providing insight into the unique topology of the system's potential energy hypersurface. With the help of this representation one is able to monitor the different explored basins of the PEL, as well as how--and mainly when--subsets of basins communicate with each other via transitions in such a way that details of the prior temporal history have been erased, i.e., local equilibration between the basins in each subset has been achieved. In this way, apart from detailed information about the structure of the PEL, the system's temporal evolution on the PEL is described. In order to gather all necessary information about the identities of two or more basins that are connected with each other, we consider two different approaches. The first one is based on consideration of the time needed for two basins to mutually equilibrate their populations according to the transition rate between them, in the absence of any effect induced by the rest of the landscape. The second approach is based on an analytical solution of the master equation that explicitly takes into account the entire explored landscape. It is shown that both approaches lead to the same result concerning the topology of the PEL and dynamical evolution on it. Moreover, a "temporal disconnectivity graph" is introduced to represent a lumped system stemming from the initial one. The lumped system is obtained via a specially designed algorithm [N. Lempesis, D. G. Tsalikis, G. C. Boulougouris, and D. N. Theodorou, J. Chem. Phys. 135, 204507 (2011)]. The temporal disconnectivity graph provides useful information about both the lumped and the initial systems, including the definition of "metabasins" as collections of basins that communicate with each other via transitions that are fast relative to the observation time. Finally, the two examined approaches are compared to an "on the fly" molecular dynamics-based algorithm [D. G. Tsalikis, N. Lempesis, G. C. Boulougouris, and D. N. Theodorou, J. Chem. Theory Comput. 6, 1307 (2010)].

On the geometrical representation of classical statistical mechanics

“…Euclidean space (EROPHILE) was initially developed in a previous work by the author regarding the dynamic response close to equilibrium 9,11 , in discrete stochastic systems whose dynamics could be described by a master equation. In that case, the driving force was to understand the underlying similarities and differences, between different dynamic relaxation computational experiments close to equilibrium, within the context of statistical mechanics for the case of a system with discrete states [12][13][14][15][16][17][18] . The result was a geometrical representation of near equilibrium dynamics where both the dynamic response and all equilibrium thermodynamic averages could be represented via Euclidean vectors.…”

“…= ∑ eq (10) In the EROPHILE representation the array of those values are transformed in Euclidean vectors ̃ via the map →̃= √ eq that results in expressing the estimation of ensemble averages as inner products. In a vector-matrix notation where observables are written as row vectors (bra-)and probabilities as column vectors (ket-) the transformation and the expectation values can be expressed in the form of Eqs (11)- (13).…”

Visualizing energy landscapes with metric disconnectivity graphs

“…Clustering landscapes by local ergodicity involves partitioning the landscape into basins about local minima. Equilibration between basins is determined by comparing forward and backward transition rates between states23 or the time-dependent probability distributions of connected basins 24…”

Computer simulations of glasses: the potential energy landscape

“…The binary Lennard-Jones mixture is one of the most commonly used model systems for glass formation, aptly described as 'the "drosophila" of computational studies of glass-forming systems' [122]. Classical forcefields are computationally cheap, and allow molecular dynamics simulations on a timescale of milliseconds (compared with nanoseconds for ab initio molecular dynamics).…”

“…A time (or equivalently temperature) axis is added to a disconnectivity graph to turn it into a 3D plot, and demonstrate the effects of aging on inter-basin communication [122].…”