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Splitting for dissipative particle dynamics. (English) Zbl 1043.60048

The author considers a system describing dissipative particle dynamics of the following form: Take \(N\) particles with position \({ q}_i\) and momenta \({ p}_i\), \(i=1,\ldots,N\), evolving in dimension \(d\), where \({ q}_i\) and \({ p}_i\) are the solutions of the SDE system \[ \begin{aligned} d{ q}_i(t)&= { p}_i dt,\\ d{ p}_i(t)&= - \sum_{j\neq i} a_{ij} V'(q_{ij}) {\hat { q}}_{ij} dt - \gamma \sum_{j\neq i} w^{\text{D}} (q_{ij}) ({\hat { q}}_{ij}\cdot { p}_{ij}){\hat { q}}_{ij} dt + \sigma \sum_{j\neq i} w^{\text{R}} (q_{ij}){\hat { q}}_{ij}\, d\beta_{ij}(t).\end{aligned} \] Here \( { p}_{ij}\) and \( { q}_{ij}\) denote relative positions and momenta and \({\hat { q}}_{ij}\) denotes the unit direction from \({ q}_{j}\) to \( { q}_{i}\) and \(q_{ij}\) the length of \({ q}_{i}\). The potential \(V\) is given by \(V(r)= \frac{1}{2}(1-\frac{r}{r_c})^2\), if \(r<r_c\) and \(0\) otherwise, where \(r_c\) is the radius of interaction. There are periodic boundary conditions prescribed on the positions \({ q}_i\) in the domain \([0,L]^d\). The parameters \(a_{ij}\) are positive and symmetric and describe the strength of repulsion between particles \(i\) and \(j\). For \(i<j\), the \(\beta_{ij}\) are IID Brownian motions and for \(i>j\), \(\beta_{ij} = - \beta_{ji}\). The functions \(w^{\text{D}}\) and \(w^{\text{R}}\) describe the dissipative and random forces, the parameters \(\gamma\) and \(\sigma\) their respective strength.
The author proposes splitting techniques to solve the system numerically, the mode of convergence discussed is weak convergence. Three numerical methods are compared: the modified Verlet method and first and second order splitting. The comparison is made with respect to the evaluation of temperature control, the velocity autocorrelation function and the radial distribution function. The final section is devoted to an explanation of the stability properties of the splitting method.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
82C80 Numerical methods of time-dependent statistical mechanics (MSC2010)
65C30 Numerical solutions to stochastic differential and integral equations
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