Random walk in energy space

N. Rathore and J. J. de Pablo (2002) Monte Carlo simulation of proteins through a random walk in energy space. J. Chem. Phys. 116, pp. 7225-7230... 

Markov Chains and Random Walks in Energy Space.601... 

In general, the extended weights are defined in a single coordinate, such as the energy, thereby projecting the random walk in configuration space to a random walk in energy space... 

For this random walk in energy space a histogram can be recorded which has the characteristic form... 

This loss of information in the projection of the random walk in configuration space has important consequences for the random walk in energy space. Most strikingly, the local diffusivity of a random walker in energy space, which for a diffusion time to can be defined as... 

For the random walk in energy space, two histograms are recorded, //+ E) and H- E), which for sufficiently long simulations converge to steady-state distributions which satisfy H E) + H E) = H E) = W E)g E). For each energy level the fraction of random walkers which have label plus is then given by fiE) = H+(E)/H(E). The above-discussed boundary conditions dictate /(E ) = 0 and f E+) = 1. 

Fig. 3. Scaling of round-trip times for a random walk in energy space sampling a flat histogram open squares) and the optimized histogram solid circles) for the two-dimensional fully frustrated Ising model. While for the multicanonical simulation a power-law slowdown of the round-trip times 0 N L ) is observed, the round-trip times for the optimized ensemble scale like 0([A ln A ] ) thereby approaching the ideal 0(A )-scaling of an unbiased Markovian random walk up to a logarithmic correction...

It should be stressed that the the random walk in state space need not be generated using this rule - in fact, for the random walk, the original Metropolis rule is usually better suited. The difference in the variance of the energy of WRMC and standard MC is ... 

Zheng, L.Q., Chen, M.G., Yang, W. Random walk in orthogonal space to achieve efficient free-energy simulation of complex systems. Proc. Natl. Acad. Sci. USA 2008,105, 20227-32. 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

L. Zheng, M. Chen, and W. Yang, Proc. Natl. Acad. Sci. U. S. A., 105(51), 20227-20232 (2008). Random Walk in Orthogonal Space to Achieve Efficient Free-Energy Simulation of Complex Systems. 

The flat distribution implies that a free one-dimensional random walk in the potential energy space is realized in this ensemble. This allows the simulation to escape from any local minimum-energy states and to sample the configurational space much more widely than the conventional canonical MC or MD methods. 

Using 112 nodes of the Earth Simulator, we performed a REMD simulation of this system with 224 replicas. The REMD simulation was successful in the sense that we observed a random walk in potential energy space, which suggests that a wide conformational space was sampled. In Fig. 4.10 we show the canonical probability distributions of the total potential energy at the corresponding 224 temperatures ranging from 250 to 700 K. 

As is clear from the Figure, all the adjacent distributions have sufficient overlaps with the neighboring ones, suggesting that this REMD simulation was successful. We indeed observed a random walk in the potential energy space. This random walk in potential energy space induced a random walk in the conformational space, and we indeed observed many occasions of the formation of native-like secondary structures (a-helix and /1-strands) during the REMD simulation. 

In this article, we have reviewed some of powerful generalized-ensemble algorithms for both Monte Carlo simulations and molecular dynamics simulations. A simulation in generalized ensemble realizes a random walk in potential energy space, alleviating the multiple-minima problem that is a common difficulty in simulations of complex systems with many degrees of freedom. 

---