^{1}and Marisol Ripoll

^{1,a)}

### Abstract

The mesoscopic simulation technique known as multiparticle collision dynamics is presented as a very appropriate method to simulate complex systems in the presence of temperature inhomogeneities. Three different methods to impose the temperature gradient are compared and characterized in the parameter landscape. Two methods include the interaction of the system with confining walls. The third method considers open boundary conditions by imposing energy fluxes. The transport of energy characterizing the thermal diffusivity is also investigated. The dependence of this transport coefficient on the method parameters and the accuracy of existing analytical theories is discussed.

The authors thank helpful discussions with Mingcheng Yang, Simone Wiegand, Andrea Costanzo, Eshan Irani, and Gerhard Gompper.

I. INTRODUCTION

II. METHOD

A. Multiparticle collision dynamics

B. Temperature profile establishment

III. BOUNDARY CONDITIONS

A. Walls with virtual particles

B. Walls with thermostats

1. Comparison of wall implementations

C. Periodic boundary conditions:Velocity exchange algorithm

IV. THERMAL DIFFUSIVITY

V. DISCUSSION AND CONCLUSIONS

### Key Topics

- Solvents
- 22.0
- Boundary value problems
- 18.0
- Thermal diffusion
- 11.0
- Hydrodynamics
- 10.0
- Particle velocity
- 7.0

##### F28

## Figures

Example of a temperature profile (circles) and the corresponding particle number density profile (squares). The symbols report the simulated values and the lines correspond respectively to Eqs. (4) and (5).

Example of a temperature profile (circles) and the corresponding particle number density profile (squares). The symbols report the simulated values and the lines correspond respectively to Eqs. (4) and (5).

(a) Temperature profile with *h* = 0.5 and walls with virtual particles. Open circles are simulation results, solid lines are the temperatures imposed at the walls, and dashed lines the actual boundary temperatures *T*(0) and *T*(*L* _{ z }). Otherwise stated the employed parameters are *h* = 0.1, α = 120, ρ = 5, a cubic size *L* = 40, *T* _{ c } = 0.9, and *T* _{ h } = 1.1. (b) Dimensionless temperature jump *T* _{ J } in Eq. (8) as a function of α (circles, down-axis), and as a function of *h* (squares, up-axis). (c) Dependence of *T* _{ J } with the inverse of the system length *L* _{ z } in the temperature gradient direction, with α = 30 and *L* _{ x } = 20 = *L* _{ y }.

(a) Temperature profile with *h* = 0.5 and walls with virtual particles. Open circles are simulation results, solid lines are the temperatures imposed at the walls, and dashed lines the actual boundary temperatures *T*(0) and *T*(*L* _{ z }). Otherwise stated the employed parameters are *h* = 0.1, α = 120, ρ = 5, a cubic size *L* = 40, *T* _{ c } = 0.9, and *T* _{ h } = 1.1. (b) Dimensionless temperature jump *T* _{ J } in Eq. (8) as a function of α (circles, down-axis), and as a function of *h* (squares, up-axis). (c) Dependence of *T* _{ J } with the inverse of the system length *L* _{ z } in the temperature gradient direction, with α = 30 and *L* _{ x } = 20 = *L* _{ y }.

*T* _{ J } for walls with thermostats as a function of α, and *h*. Symbols and parameters similar to Fig. 2. The inset shows the detail of a temperature profile close to the wall for with α = 30, *h* = 0.1, ρ = 5, and *L* _{ z } = 40.

*T* _{ J } for walls with thermostats as a function of α, and *h*. Symbols and parameters similar to Fig. 2. The inset shows the detail of a temperature profile close to the wall for with α = 30, *h* = 0.1, ρ = 5, and *L* _{ z } = 40.

Illustration of the periodic simulation box in the presence of a temperature gradient.

Illustration of the periodic simulation box in the presence of a temperature gradient.

(a) Temperature profile obtained from the velocity exchange algorithm with *h* = 0.1, α = 120, and ρ = 10. Symbols correspond to the measured temperatures, dashed-line is the estimated temperature profile from Eq. (10). (b) Velocity squared distribution in Eq. (9) for the temperatures *T* _{ c } = 0.9 and *T* _{ h } = 1.1, typically used for the cold and hot baths.

(a) Temperature profile obtained from the velocity exchange algorithm with *h* = 0.1, α = 120, and ρ = 10. Symbols correspond to the measured temperatures, dashed-line is the estimated temperature profile from Eq. (10). (b) Velocity squared distribution in Eq. (9) for the temperatures *T* _{ c } = 0.9 and *T* _{ h } = 1.1, typically used for the cold and hot baths.

Thermal diffusivity for two values of ρ. (a) *k* _{ T } as a function of α for *h* = 0.1. (b) *k* _{ T } as a function of *h* for α = 120. The insets are a zoom-in for large values of α and small *h*, respectively. Lines correspond to the analytical approach in Eq. (11) and symbols to simulation results. Continuous lines correspond to ρ = 5 and dashes lines to ρ = 20. Triangles refer to simulations with walls and thermostats, circles to walls with virtual particles, and squares to the velocity exchange algorithm.

Thermal diffusivity for two values of ρ. (a) *k* _{ T } as a function of α for *h* = 0.1. (b) *k* _{ T } as a function of *h* for α = 120. The insets are a zoom-in for large values of α and small *h*, respectively. Lines correspond to the analytical approach in Eq. (11) and symbols to simulation results. Continuous lines correspond to ρ = 5 and dashes lines to ρ = 20. Triangles refer to simulations with walls and thermostats, circles to walls with virtual particles, and squares to the velocity exchange algorithm.

Relative deviation of the simulated thermal diffusivity *k* _{ T, sim } with respect to the analytical approach *k* _{ T, an }, Δ*k* _{ T }/*k* _{ T } ≡ (*k* _{ T, sim } − *k* _{ T, an })/*k* _{ T, an } a) as a function of α (b) as a function of *h*. Simulation values are those presented in Fig. 6 for walls and thermostats. Open symbols and dashed lines employ both analytical contributions in Eq. (11), while solid symbols and solid lines take into account the collisional contribution in Eq. (15).

Relative deviation of the simulated thermal diffusivity *k* _{ T, sim } with respect to the analytical approach *k* _{ T, an }, Δ*k* _{ T }/*k* _{ T } ≡ (*k* _{ T, sim } − *k* _{ T, an })/*k* _{ T, an } a) as a function of α (b) as a function of *h*. Simulation values are those presented in Fig. 6 for walls and thermostats. Open symbols and dashed lines employ both analytical contributions in Eq. (11), while solid symbols and solid lines take into account the collisional contribution in Eq. (15).

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