Markovian dynamics

An essential property of A is the existence of the inverse transformation A . This allows us to go back and forth between Hamiltonian dynamics and Markovian dynamics. In other words, A maps deterministic reversible dynamics to irreversible stochastic dynamics. 

A cornerstone of condensed phase reaction theory is the Kramers-Grote-Hynes theory. In a seminal paper Kramers solved the Fokker-Plank equation in two limiting cases, for high and low friction, by assiuning Markovian dynamics y(t) 5(t). He foimd that the rate is a non-monotonic function of the friction ( Kramers turnover .) Further progress was made by Grote and Hynes - who... 

An earlier approacfr was to solve the quantum problem in the high-temperatme limit using Markovian dynamics and assuming a parabolic barrier. The quantum rate has the following formF - ... 

To make further progress, it is standard practice to take this definition of the spectral density and replace it by a continuous form based on physical intuition. A form that is often used for the spectral density is a product of ohmic dissipation qco (which corresponds to Markovian dynamics) times an exponential cutoff (which reflects the fact that frequencies of the normal modes of a finite system have an upper cutoff) ... 

Extensions of the Kramers model are considered necessary [92-94, 97-99] although there are refined versions of the original formulation [100, 101]. Such non-Markovian dynamics has been taken into consideration... 

Hamm P, Lim M, Hochstrasser RM. Non-Markovian dynamics of the vibrations of ions in water from femtosecond infrared three pulse photon echoes. Phys Rev Lett 1998 81 5326-5329. 

The Markovian limit corresponds to Xi - oo. By solving Eq. (73) to the lowest order in Xf it is easy to see that in this case the non-Markovian dynamics leads to an enhancement of the decay rate k. In the notation of ref. 22b, an approximate expression for Eq. (69) may then be written as... 

What is the significance of the Markovian property of a physical process Note that the Newton equations of motion as well as the time-dependent Schrodinger equation are Markovian in the sense that the future evolution of a system described by these equations is fully determined by the present ( initial ) state of the system. Non-Markovian dynamics results from reduction procedures used in order to focus on a relevant subsystem as discussed in Section 7.2, the same procedures that led us to consider stochastic time evolution. To see this consider a universe described by two variables, zi and z, which satisfy the Markovian equations of motion... 

In essence, the DPS approach reduces the problem of global kinetics to a discrete space of stationary points. Phenomenological rate constants can then be extracted under the assumption of Markovian dynamics within this space, which requires that the system has time to equilibrate between transitions and lose any memory of how it reached the current minimum. The Markovian assumption is therefore an essential part of the framework. However, we can regroup the stationary points into states whose members are separated by low barriers so that the Markov property is likely to be better obeyed between the groups (Section 14.2.3). 

The metadynamics method was introduced in 2002 by Laio and Parrinello as an elegant extension of adaptive bias potential methods [65]. The authors used a coarse-grained non-Markovian dynamics in the space defined by a few collective coordinates s,. With the aid of a history-dependent potential term the minima of the FES were filled in time, allowing the efficient exploration and accurate determination of the FES as a function of the collective coordinates. Laio and Parrinello demonstrated the appUcabUity of this approach in the case of the dissociation of a sodium chloride molecule in water and in the study of the conformational changes of a dialanine in solution [65]. 

For metallic environments, non-adiabatic effects in the form of electron-hole pair coupling is known to dominate the dissipative dy-namics. The bath of electron-hole pairs typically thermalizes within a few femtoseconds, which allows for a great simplification of the memoiy kernel. In view of the adsorbate dynamics, the bath thus remains in its equilibrium state and the memoiy kernel can be treated in the second-order Born-Markov approximation. For Markovian dynamics, the dissipative Liouvillian is often written in its diagonal Lindblad form " ... 

The Andersen thermostat is very simple. After each time step Si, each monomer experiences a random collision with a fictitious heat-bath particle with a collision probability / coll = vSt, where v is the collision frequency. If the collisions are assumed to be uncorrelated events, the collision probability at any time t is Poissonian,pcoll(v, f) = v exp(—vi). In the event of a collision, each component of the velocity of the hit particle is changed according to the Maxwell-Boltzmann distribution p(v,)= exp(—wv /2k T)/ /Inmk T (i = 1,2,3). The width of this Gaussian distribution is determined by the canonical temperature. Each monomer behaves like a Brownian particle under the influence of the forces exerted on it by other particles and external fields. In the limit i —> oo, the phase-space trajectory will have covered the complete accessible phase-space, which is sampled in accordance with Boltzmann statistics. Andersen dynamics resembles Markovian dynamics described in the context of Monte Carlo methods and, in fact, from a statistical mechanics point of view, it reminds us of the Metropolis Monte Carlo method. 

By taking the time derivative of this integral, one obtains a set of coupled equations for the density matrix of the system Ps t) and the n auxiliary matrices Pi(t). The main point is that this technique allows us to carry out non-Markovian dynamics by a system of coupled equations local in time. 

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