Markov Model Theory
This section reviews the relation between the continuous dynamics of a molecular system in thermal equilibrium and the kinetics given by a Markov State Model (MSM). We will introduce the dynamical propagator, an error-less, alternative description of the
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Markov Model Theory Marco Sarich, Jan-Hendrik Prinz, and Christof Schütte
3.1
Continuous Molecular Dynamics
A variety of simulation models that all yield the same stationary properties, but have different dynamical behaviors, are available to study a given molecular model. The choice of the dynamical model must therefore be guided by both a desire to mimic the relevant physics for the system of interest (such as whether the system is allowed to exchange energy with an external heat bath during the course of dynamical evolution), balanced with computational convenience (e.g. the use of a stochastic thermostat in place of explicitly simulating a large external reservoir) [8]. Going into the details of these models is beyond the scope of the present study, and therefore we will simply state the minimal physical properties that we expect the dynamical model to obey. In the following we pursue the theoretical outline from Ref. [31] (Sects. 3.1–3.7) and Ref. [37] (Sects. 3.1–3.8) which should both be used for reference purposes. Consider a state space Ω which contains all dynamical variables needed to describe the instantaneous state of the system. Ω may be discrete or continuous, and we treat the more general continuous case here. For molecular systems, Ω usually contains both positions and velocities M. Sarich · J.-H. Prinz · C. Schütte (B) Freie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany e-mail: [email protected]
of the species of interest and surrounding bath particles. x(t) ∈ Ω will denote the state of the system at time t. The dynamical process considered is (x(t))t∈T , T ⊂ R0+ , which is continuous in space, and may be either time-continuous (for theoretical investigations) or time-discrete (when considering time-stepping schemes for computational purposes). For the rest of the article, the dynamical process will also be denoted by x(t) for the sake of simplicity; we assume that x(t) has the following properties: 1. x(t) is a Markov process in the full state space Ω, i.e. the instantaneous change of the system (dx(t)/dt in time-continuous dynamics and x(t + t) in time-discrete dynamics with time step t), is calculated based on x(t) alone and does not require the previous history. In addition, we assume that the process is time-homogeneous, such that the transition probability density p(x, y; τ ) for x, y ∈ Ω and τ ∈ R0+ is well-defined: p(x, A; τ ) = P x(t + τ ) ∈ A x(t) = x (3.1) i.e. the probability that a trajectory started at time t from the point x ∈ Ω will be in set A at time t + τ . Such a transition probability density for the diffusion process in a onedimensional potential is depicted in Fig. 3.1b. Whenever p(x, A; τ ) has an absolutely continuous probability density p(x, y; τ ) it is given by integrating the transition probability density over region A:
G.R. Bowman et al. (eds.), An Introduction to Markov State Models and Their Application to Long Timescale Molecular Simulation, Advances in Experimental Medicine and Biology 797, DOI 10.1007/978-94-007-7606-7_3, ©
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