Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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Nose-Hoover thermostat Atomic velocities establish a thermometer, as argued in Chapter 6.2.4. All algorithms performing the role of thermostat use some modifications of Newton’s second law of motion to provide a constant average temperature of the particle ensemble by adding and removing its energy [55]. The temperature control is illustrated here by the Nose-Hoover algorithm as formulated by Holian and co-authors [39]. The Nose-Hoover thermostatted MD has a theoretical basis in the classical thermody namic concept of "connecting" a thermodynamic system (in this case, the MD cell) with a heat reservoir that ensures a constant temperature during the simulation. Consequently, the given atomic ensemble becomes, by definition, canonical ( N , V , T ). The Nose-Hoover model uses the classical temperature concept (discussed in Chapter 6.2.4) based on the instantaneous kinetic energy of the system. The technique of providing thermostatic condi tions is of the integral-feedback type. First, an additional term is inserted into Newton’s second law of motion to control the „jigllings and wigllings“ of atoms. Second, the dynamic variable, ξ H , must satisfy the additional equation of motion ˙ ξ H = ϑ H T T 0 − 1 (6.20) which provides the necessary feedback. In the differential Equation (6.19), ϑ H is the coupling speed of the atom with the thermal reservoir. The standard form of Newton’s equation of motion is recovered from expression (6.19) for ϑ H = 0. The variation of the heat distribution variable ξ H ensures that the long-term average kinetic energy (consequently, the average temperature as well) remains constant, while allowing fluctuations in its current value (Figure 6.10). Numerical integration of Nose-Hoover equations of motion using the Störmer central difference algorithm ((6.10) and (6.11)) [56] reduces to the following expressions r ( t ) = r ( t − δ t )+ δ t ˙ r ( t − δ t / 2 )+ O ( δ t 3 ) , ξ H ( t ) = ξ H ( t − δ t )+ ϑ H T ( t − δ t / 2 ) T 0 − 1 δ t + O ( δ t 3 ) , (6.21) ˙ r t + δ t 2 = 1 1 + ϑ H ξ H ( t ) δ t / 2 ˙ r t − δ t 2 1 − 1 2 δ t ϑ H ξ H ( t ) + F ( t ) m δ t + O ( δ t 3 ) . ¨ r i = F i m i − ϑ H ξ H ˙ r i (6.19)

A detailed consideration of this method is available in [39]. A more advanced version of the thermostat is available in [57].