Thermostat Andersen

Another method of controlling the temperature that can be used in CP MD is the stochastic thermostat of Andersen.27 In this approach the velocity of randomly selected nucleus is rescaled this corresponds in a way to the stochastic collisions with other particles in the system. Therefore, this approach is often called a stochastic collision method. The Andersen thermostat has recently been shown28 to perform very well in the Car-Parinello molecular dynamic simulations of bimolecular chemical reactions. 

A full description of the outline and the capabilities of the NEWTON-X package is given elsewhere [38], In brief, the nuclear motion is represented by classical trajectories computed by numerical integration of Newton s equations using the Velocity-Verlet algorithm [39], Temperature influence can be added by means of the Andersen thermostat [40], The molecule is considered to be in some specific... 

If these equations are equipped with an appropriate thermostat, for example, a stochastic Andersen thermostat [53] or a deterministic Nose-Hoover thermostat [37], the resulting distribution in path space is consistent with exp( = ab[x( ]- Alternatively, a stochastic Lan-... 

In a KMC method, it is typically assumed that various possible state-to-state transitions from a given state are well modelled by the Arrenhius law and then molecular dynamics is used to calculate the prefactor A and energy difference AE in order to understand the timescales and relative probabilities of different rare events. A Markov state model can be developed to help understand the global dynamics and simplify the model as a whole. For references on many interesting approaches to this important topic, the reader is referred to [36,42,137,149,391]. Andersen Thermostat. Of particular interest is the simple and useful Andersen thermostat [11]. This method works by selecting atoms at random and randomly perturbing their momenta in a way consistent with prescribed thermodynamic conditions. It has been rigorously proven to sample the canonical distribution [114],... 

In the next subsection we describe the numerical treatment of the DPD system. We then take up the alternative scheme, the pairwise Nos6-Hoover-Langevin (PNHL) method, of [228] which also conserves momentum exactly. Other alternatives to the DPD system include the Peters thermostat [299], the Lowe-Andersen method [241], and the Nos6-Hoover-Lowe-Andersen thermostat [347]. We do not discuss these schemes here, but note that they are compared to DPD and some of the other schemes introduced below in [228]. 

The equation of motion for the Andersen thermostat can formally be written... 

It can be shown [55] that the Andersen thermostat with non-zero collision frequency a leads to a canonical distribution of microstates. The proof [55] involves similar arguments as the derivation of the probability distribution generated by the MC procedure. It is based on the fact that the Andersen algorithm generates a Markov chain of microstates in phase space. The only required assumption is that every microstate is accessible from every other one within a finite time (ergodicity). Note... 

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. 

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