# Interdisciplinary Applied Mathematics

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P(t; Г) = Гехр[—Tt],

where P(t; Г) dt is the probability that the next collision will take place in the interval [t, t + dt].

A constant-temperature simulation employing Andersen thermostat consists of the following steps:

1. Start with an initial set of positions and momenta and integrate the equations of motion for a time interval of At.

2.    A    number    of    particles    are selected    to undergo    a    collision    with    the

heat bath. The probability that a particle is selected in a time step of length At is TAt.

3. If particle i has been selected to undergo a collision, its new velocity will be drawn from a Maxwell-Boltzmann distribution corresponding to the desired temperature T0. All other particles are unaffected by this collision.

It should be noted that rigorously, the dynamics generated by the Andersen scheme are unphysical (Frenkel and Smit, 2002). Therefore, it is risky to use the Andersen method when studying dynamical properties. The key properties of the Andersen thermostat can be summarized as follows:

1.    It    relies    on    stochastic    collisions    with    heat reservoirs    to    control    the

temperature.

2. It produces a canonical NVT distribution, but probabilistically, rather than deterministically.

3. The algorithm is very similar to MD, but each particle undergoes a stochastic collision with probability TAt after every time step.

The Andersen thermostat has been used very effectively in dissipative particle dynamics (DPD) methods; see Section 16.4 in this chapter.

Table 16.2 gives a comparison of the temperature coupling schemes that are commonly used in MD simulation.

###### 16.1.4 Data Analysis

MD simulation generates the trajectories of all the particles in the system, but to obtain a deeper insight into the system being studied, we need to analyze the trajectories obtained during the MD simulation. In this section, we summarize some of the most commonly performed data analyses in the simulation of fluid transport.

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