Mathematical Physics - Volume II - Numerical Methods

6.2 Molecular Dynamics

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The strain tensor components are calculated based on the deviation of the current network configuration from the reference configuration defined at the initial moment. Since atomic positions are known at all times (therefore, in both the initial and the current configuration), calculating the strain components is a straightforward task. For example, the components of the left Cauchy-Green strain tensor, corresponding to i -th atom of the plane system, are commonly (e.g., [37]) defined as follows

6 ∑ j = 1 ( ¯ r i j ) α ( ¯ r i j ) β , ( ¯ r i j ) α = ( r i j ) α / r 0 .

1 3

b αβ =

(6.15)

It should be noted that, unlike the stress definition (6.13), the virial equation (6.15) is instantaneously valid in time and space, that is, it does not require time averaging. Temperature The temperature evaluation during dynamic processes raises some fundamental questions related to the thermodynamics of nonequilibrium processes in relation to basic thermostatic concepts such as entropy and absolute temperature of nonequilibrated systems (e.g., [34], [38], [39]). That discussion is bypassed herein. It is deemed sufficient just to point out the Callen’s [40] claim that the nonequilibrium definitions of entropy and temperature are based on uncertain premises. Be it as it may, a consensus has been reached, over time, on the use of a standard definition of temperature, known from the kinetic theory of gases. This definition is based on the equipartition theorem that provides a relationship between the average kinetic energy and the instantaneous kinetic temperature of the system: atomic velocities establish a thermometer. According to this theorem, each degree of freedom contributes k B T / 2 to the internal energy of the system from where it follows T ≡ 2 3 k B ⟨ E k ⟩ = 2 3 k B mv 2 2 . (6.16) where k B is the Boltzmann constant. The definition (6.16) is firmly established in statistical mechanics as it derives from the distribution of the highest probability of a canonical ( N , V , T ) ensemble [34], [41]. Thus, expression (6.16) tacitly implies that the system is thermally equilibrated, and the atomic velocities distributed in accordance with the Maxwell-Boltzmann distribution (6.17) [42]. As argued by Holian and coworkers [39], the instantaneous kinetic temperature is the only meaningful definition in nonequilibrium situations. Importantly, the intensity of velocity vector v ( v x , v y , v z ) appearing in Equation (6.16) corresponds to the vibratory motion and, therefore, does not contribute to the resultant momentum. Nonetheless, it possesses a finite kinetic energy that is identified with the thermal energy and is related to the absolute temperature in MD simulations. With reference to Figure 6.6a, the vibrational velocity components could be obtained by subtracting the velocity of correlative (systematic) motion (full black line) from the total (individual) atomic velocity (dashed red line). The total velocity of each atom is obtained directly by solving Newton’s equations of motion (6.1) 1 . The associated velocity of correlative motion can be calculated by spatially averaging the total velocities of all atoms belonging to a particular averaging region centered at the atom in question (Figure 6.6b). This averaging