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Potential Energy Landscapes

By Peyman Khorsand

The configuration space of any system with conserved energy (time-independent Hamiltonian) can be broken into subspaces surrounding potential energy minima, which the minima will be reached by the steepest descent path of the potential. This hyper-volume around a potential minimum is called basin. The structure based on these neighboring basins is called as the “Inherent Structure”. The statistical behavior of a complex system can be studied by analyzing the characteristics of this structure. The partition function of a system of N particles can be written as

Z(\Omega,N,T) = {1\over N! \lambda^{3N}} \int_{\Omega} d^{3N}x e^{-\beta V(x)}

We characterize each basin by its minimum potential E_i. The total volume \Omega is partitioned by these basins and so does the above integral.

Z(\Omega,N,T) ={1\over N! \lambda^{3N}} \sum_i \int_{\omega_i} d^{3N}x e^{-\beta V(x)}

Defining an average free energy for basins f_{b}(E_i,T,\Omega)

-\beta f_{b}(E_i,T,\Omega)=\ln \langle {\int_{\omega_i} d^{3N}x e^{-\beta V(x)} \over \lambda^{3N}} \rangle

Now if we know the probability distribution of energy minima \rho(E) the partition function can be written easily as,

Z(\Omega,N,T)=\int dE \rho(E) e^{-\beta f_{b}(E,T,\Omega)}

General Properties of PEL’s:

The number of total minima is exponentially depends on the number of underlying degrees of freedom (e.g. number of atoms),

N_{min} \sim e^{N_{deg}}

The energy distribution of minima follows Guassian distribution. This can be understood through central limit theorem for a large system with independent subsystems.

P(E) = {\cal{N}} e^{-(E-E_{0})^2/2 \sigma_e^2}

Also on average an exponential relation between the minimas’ energy and their corresponding volume was observed

A \sim e^{-\alpha E}

The probability distributions, P(A), of the volume, A, of these basins for various dynamical systems (Binary Lennard-Jones, Dzugutov Liquids and Amorphous Silicon) show power law behavior

P(A) \sim A^{-2}

There is a similarity between this phenomenon and “Apollonian Packing” of an arbitrary volume, where in the limit of large dimensionality of the target space obeys the same power law. Moreover, the network consist of connecting neighboring minima is a scale free network, which means

There is a strong correlation (a power law) between the volume of the basin and its number of neighbors, k_i,

k_i \sim (A_i)^\beta

This relation subsequently dictates a relation between the minimum energy and number of neighbors.

As a reference look through the papers by J. P. K. Doye and C. P. Massen.

This entry was posted on October 14, 2008 at 12:38 am and is filed under Complex Systems Dynamics. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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