Stochastic thermostatics

P. Gueret, P. Merat, M. Moles, and J. P. Vigier, Stable states of a relativistic harmonic oscillator imbedded in a random stochastic thermostat, Lett. Math. Phys. 3(1), 47-56 (1979). 

Sindhikara DJ, Kim S, Voter AF, Roitberg A (2009) Bad seeds sprout perilous dynamics stochastic thermostat induced trajectory synchronization in biomolecules. J Chem Theory Comput 5(6)4624-1631... 

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. 

The additivity property of thermostats mentioned above holds also for stochastic thermostats, since the Fokker-Planck equation is constructed from linear operations on the vector fields involved, thus it is enough to require that the newly introduced terms satisfy appropriate fluctuation-dissipation relationships so that they are compatible with the extended Gibbs distribution of the deterministic part. That is, we require... 

Davidchack, R. Discretization errors in molecular dynamics simulations with deterministic and stochastic thermostats. J. Comput. Phys. 229, 9323-9346 (2010). doi 10.1016/j.jcp.2010. 09.004... 

Leimkuhler, B., Noorizadeh, E., TheU, E A gentle stochastic thermostat for molecular dynamics. J. Stat. Phys. 135, 261-277 (2009). doi 10.1007/sl0955-009-9734-0... 

It was logical to extend the results of the thermostatic theory to temporal changes. Stochastic thermostatics adopts an intermediate level between statistical and phenomenological thermodynamics. Analogously, in principle the stochastic treatment of thermodynamics processes has an intermediate character between nonequilibrium statistical mechanics and phenomenolog-... 

In the simnlations of Hecht et al. [16], the simple coUisional coupling procedure described in Sect 7.1 was used. This means that the colloids were treated as point particles, and solvent particles could flow right through them. Hydrodynamic interactions were therefore only resolved in an average sense, which is acceptable for studies of the general properties of an ensemble of many colloids. The heat from viscous heating was removed using the stochastic thermostat described in Sect. 2.3. 

Today a large variety of methods exist that drive the system into the canonic state, e.g., by introduction of artificial degrees of freedom or by coupling the system to a heat bath via stochastic methods. The reader will find more details in Refs. 12 and 13. The choice for the present work is a Langevin thermostat [14], This means that instead of integrating Newton s equations of motion, one solves a set of Langevin equations,... 

More realistic kinetic behavior in implicit solvent simulations can be obtained with the Langevin thermostat [18] where stochastic collisions and friction forces provide kinetic energy transfer to and from the solute in an analogous fashion to explicit solute-solvent interactions. As a result, kinetic transition rates similar to rates from explicit solvent simulations can be recovered with an appropriate choice of the friction constant [2]. 

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 this section, we consider the combination of stochastic perturbation with a deterministic thermostat. Methods constructed in this way can be ergodic for the canonical distribution while also providing flexibility in way equilibrium is achieved. We distinguish in (6.16) between multiplicative noise, where B = B(z) varies with z and additive noise, where B is constant. In our treatment of this topic we will only consider additive noise. The presence of multiplicative noise may complicate discretization. As we shall see, the reliance on additive noise improves the performance of discretization schemes. 

The Nose-Hoover-Langevin (NHL) method is based on a simple idea replace the chain in the Nos6-Hoover Chain, whose sole purpose is to maintain a Gaussian distribution in the auxiliary variable, by a stochastic Langevin-type thermostat. The method was first proposed in [323]. The proof of ergodicity (more precisely the confirmation of the Hormander condition), for a problem with harmonic internal interactions, was given in [226] and we roughly follow the treatment from this paper. 

An alternative stochastic-dynamical method for thermostatting molecular dynamics is the stochastic velocity rescaling method proposed by Bussi and collaborators [60, 63] and somewhat generalized in [225]. The equations are (for the case Af = /, considered for simplicity) ... 

We set >T = yp = y and compare simulations using three values of y. Choosing y = 0 removes any stochastic terms from the simulation, reducing the dynamics to a similar form to the Nos6-Hoover thermostat. The distribution of the pressure FI, plotted in Fig. 8.7 (centered on the target pressure P), is well sampled under all three choices of the friction y. However, as in the NVT experiment in Fig. 8.2, without any stochastic terms in the equations of motion the system exhibits a pronounced... 

The other class of fluctuation phenomena, well-known since the work of Gibbs (done in 1902, see Gibbs (1948)) and Einstein (1910), is equilibrium fluctuation. The theory of equilibrium (thermostatic) fluctuations considers the equilibrium state as a stationary stochastic process (see, for example, Tisza Quay (1963) and Tisza (1966)). By thermostatic fluctuation theory the statistical character (e.g. the distribution functions and moments derived from it) can be computed. 

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