Computational biochemistry ferenc Bogár György Ferency
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- 3. Statistical mechanics background
- 3.2. Ensembles: NPT, NVT, micro canonical, canonical
- 3.3. Probability distribution in microcanonical, canonical ensembles
- 3.4. Calculation of ensemble averages
Velocity-Verlet [3] The position vector after the time step can be directly calculated from the Taylor expansion (first equation of (6.2)). Substituting the acceleration with the forces, using the Newton’s equation
To obtain this we need the position, velocity and force values at tk. But for the next step we also need the velocity at tk that time, vi(tk + Δt). This can be calculated as
Here we used linear approximation of the acceleration in the time interval of (tk, tk + Δt). Using again the Newton’s equation we obtain:
The steps of the solution of these equations can be seen in Figure 6.5. 
Figure 6.5. Steps of the velocity-Verlet integration algorithm 3. Statistical mechanics background 3.1. Microstates, macrostates An extended molecular system (e.g. protein in a solvent) has a very large number of degrees of freedom. Its states in classical mechanics can be described by the coordinates (r) and momenta (p) of the N particles the system consists of. Every possible combination of these vectors describes a microstate (r1, r2, . . ., rN, p1, p2, . . ., pN). The possible microstates form together the phase space. A probability distribution P(r1, r2, . . ., rN, p1, p2, . . ., pN) of the microstates defines a statistical ensemble. In biomolecular simulations we only use ensembles related to an equilibrium state of the system under given macroscopic environmental constraints (e.g. fixed volume or pressure). In statistical physics our goal is to derive the macroscopic properties (macrostates) of a complex physical system, knowing the particles as well as their interactions in it . To reach our goal we have to know theoretically all of the microstates of the system together with their probabilities (i.e .P). In general it is inevitable to know all these data, in most cases we have to settle for the collection of a representative sample that gives a good approximation of the total ensemble. The central problem of the simulation is, how we can produce a proper approximation of an ensemble using a limited sampling time. Before discussing this problem, we have to mention the problem of ergodicity. Our system is ergodic if a single copy of the system will go through all of its microstates, if we follow its evolution (trajectory) in the state space for an appropriately long time. Unfortunately this time can also be infinite. In practical simulations we have to find a proper sampling of the ensemble that provides approximate values for the macrostates (averages) which are close enough to the exact values. One of key questions of the simulation is: How can we test the accuracy of this approximation if we do not know the exact values? What can be done is to calculate the value of a selected property and test its convergence as the simulation time grows (of course, this can also be problematic, because nothing guarantees that the convergence is uniform). 3.2. Ensembles: NPT, NVT, micro canonical, canonical The probabilities of the microstates depend on the macrostate of the system, which is determined (according to the classical thermodynamics) by the thermodynamic variables. The most often used conjugate variable pairs are the entropy/temperature (S/T), volume/pressure (V/p), particle number/chemical potential (N,μ). The first members of these pairs are extensive, while the second ones are intensive quantities. Fixing any member of the three conjugate pairs we obtain a specific ensemble. The most often used ones are the NVE (microcanonical), NVT (canonical) and NPT (isotherm-isobar) ensembles. The thermodynamic state of these ensembles are defined by fixing quantities listed in the name of the ensemble (e.g. N: particle number, V: volume and E: energy). 3.3. Probability distribution in microcanonical, canonical ensembles In a microcanonical ensemble the system is isolated from its environment, neither material nor energy transport is allowed. In this case each microstate has equal probability. If our system, with fixed particle number and volume, is in equilibrium with a heath bath (which allows the energy exchange between the system and its environment) it is termed as canonical ensemble. Its states follow the Maxwell-Boltzmann statistics that is the probability of finding the system in a dV=(dr,dp) volume element at (r,p)= (r1, r2, . . ., rN, p1, p2, . . ., pN) in the state space is
where U(r) in the potential energy of the system,
is the partition function. 3.4. Calculation of ensemble averages In statistical physics the average of physical quantities f(r,p) can be calculated using the above defined probabilities:
We mention here an alternative formulation of the ergodicity: the ensemble average of an arbitrary quantity is equal to its time average, i.e.
where r(τ) and p(τ) denote the positions and momenta of the atoms at the time τ, respectively. Averaging using trajectories from equilibrium MD simulations Having proper sample of the statistical ensemble of our system the averages can be calculated directly. From the position and momentum vectors we can calculate the value of fi (r,p) the physical quantity f at the i-th time point of the trajectory. If we have altogether M points along the trajectory the average is
The standard deviation of the quantity is
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