Simulating thermal fluctuations in soft matter models.

The structure of the thesis is organized as follows: In chapter 2, a brief introduction to molecular dynamics is given. Chapter 5 presents a brief summary of the research work and the main conclusions highlighted from the two different cases examined in this thesis, and discusses possible future work.

Introduction

Using such thermostats, MD simulations can be performed under NVT [26] (constant number of particles, volume and temperature) ensembles or NPT [27] (constant number of particles, pressure and temperature) ensembles. This thesis deals with MD simulations performed under constant temperature (NVT ensemble).

Molecular Dynamics ensembles

Microcanonical ensemble

The delta of the Dirac function mathematically forces energy conservation within this ensemble, since it ensures that δ(H−E) 6= 0 when H =E. However, this proves to be extremely difficult to evaluate, since ensemble averages (see equation 2.2.1) require knowledge of all possible microscopic states.

Canonical ensemble

Using the ergodic hypothesis, ensemble averages are equivalent to time averages (see Equation 2.2.2)) obtained by performing experiments numerically during MD simulation. Using the canonical distribution function, given by Eq. 2.2.6), one can easily show how to obtain the famous Maxwell distribution function.

Estimating temperature in MD simulations

2.2.9), states that the equilibrium average of the kinetic energy of any particle is constant and equal to kBT /2.

Basic Approach to MD Simulation

Reduced Units

Starting with a set of initial conditions for the phase space coordinates of the system, new phase space coordinates are calculated at each iteration within the MD loop. In addition, reduced units are generally preferred when running computer simulations to preserve the accuracy of the results by avoiding the rounding associated with arithmetic calculations between very large and small numbers, since the values ​​are approximately the same.

Figure 2.1: A figure showing typical algorithmic steps for a molecular dynamics simu- simu-lation

Analysis of Simulation Data

Radial Distribution Function

For a fixed density, the radial distribution function g(r) defines the probability of finding a particle at a distance from a reference particle compared to the expected probability of such a particle in a completely random distribution such as the ideal gas state. Furthermore, for a system of interacting particles, the thermodynamic properties such as energy and pressure can be calculated from the g(r), using the general relationship,.

Diffusion Coefficient

The temperature value can be calculated from the fluctuation velocities ci of the particles according to the function of Eq. Thermostatic schemes based on the flow rate assumption are known as the "profile biased" thermostat (PBT).

Nosé-Hoover Thermostat

Our work is based on the original formulation of the configurative thermostat presented by Braga and Travis. However, it is worth giving a brief introduction to the Nosé-Hoover thermostat since the work of Braga and Travis is in the spirit of the Nosé-Hoover thermostat.

Braga-Travis Thermostat

As previously mentioned, the B-T thermostat is easier to implement than previously derived thermostat configuration schemes such as those presented by Delhommelle-Evans [37]. Using the phase space formulation, we can integrate the equations of motion using the reversible symmetric Trotter algorithm, which we will call STP.

Configurational Temperature Nosé-Hoover thermostat

However, there are problems associated with this thermostat, such as non-nergodicity for rigid systems, and it is difficult to derive a general formalism of the equations of motion without losing the characteristically conserved nature of the Hamiltonian. M⇣ is the mass of the thermostat, kB is the Boltzmann constant and T is the temperature of the thermal bath. Using the operator formula as shown in Appendix (A.1), the propagator action on Eq.

In the case of a general potential, the application of the operator formula becomes difficult unless the calculated force is linearly approximated around a particular position, e.g. In such a case, by applying the operator formula, the effect of the propagator on Eq. 3.4.16) for a general continuous potential can be approximated as,. However, by fitting a standard Nosé-Hoover thermostat, which effectively regulates the kinetic energy of the configured thermostat, one can create a chain of thermostats capable of producing an ergodic system for such rigid systems.

CTNH Dynamics in a WCA Potential

It can be easily shown that the second partial derivative of the potential with respect to x is given by the following expression, In our work, simulations of the CTNH thermostat were performed using the WCA potential with the PDHA-STP algorithm. 3.1) shows the normalized conserved volume of the CTNH thermostat and the Nosé-Hoover thermostats.

It is worth noting that the error introduced by the linearization of the forces is n 1%. The percent error is calculated based on a first order approximation of the forces and linearly approximated forces along the x, y and z directions. The simulation times for the CTNH thermostat were also compared with the standard NH thermostat and it was found that the NH thermostat was 10% faster than the run times of the CTNH thermostat.

Figure 3.1: Comparison of the normalized hamiltonian function for the WCA potential function coupled to a configurational temperature Nosé-Hoover thermostat shown by the filled circles ( ) and the Nosé-Hoover thermostat shown by the filled triangles ( N )

Conclusion

The GCM potential is finite which means that a complete superposition of particles is possible resulting in a number of unusual properties appearing in the solid and liquid phases. Examining the phase diagram of the GCM as shown in Fig.(4.1), we can observe unusual features that are present in the phase diagram of water. GCM re-entry melting can be observed from the phase diagram when the system is isothermally compressed from state P to Q, it undergoes two state changes.

Also, at a sufficiently low temperature the system can exhibit different crystalline structures depending on the density; the symmetry of the structure can be either face- or body-centered [47]. In this work, the shape of the potential for the GCM is expressed as follows. The aim of our study is to investigate the significance of the zero-point effects on the GCM within the range of temperatures and densities explored by Mausbach and May [17].

Figure 4.1: Gaussian-Core phase diagram [47]

Phonostat Methodology

• Normal Coordinate Analysis
• Normal mode calculation
• Modal Temperature using the Wigner distribution function
• Modal Temperature control using a Nosé-Hoover Chain thermostat 38

The problem we will face is how to evaluate the elements of the Hessian matrix. It is also possible to recover Cartesian phase space coordinates by projecting the phase space coordinates of the normal mode onto the inverse matrix of the normal modes. Classically, from equivalence theory, the temperature of each mode is the same regardless of the mode oscillation frequency.

The temperature of each mode depends on the vibrational frequency of the mode as expressed by Eq. The generalized form of the equations of motion for the Nosé-Hoover chain can be expressed in the simple form as Using the direct translation technique, the pseudo-code form of the time-reversible algorithm is:

Numerical Simulation

By parity theory, classically, the temperature of each mode is the same regardless of the mode's vibrational frequency. Table (4.2) shows the numerical pressure values ​​and the corresponding percentage difference obtained from classical MD simulations using Cartesian coordinates and classical MD simulation using the phonostat algorithm for different temperature values ​​investigated by Mausbach and May [17]. This comparison highlights the level of accuracy of the phonostat algorithm used in reproducing classical results, in this case, when comparing thermodynamic properties such as pressure.

Furthermore, a comparison of the static properties is also shown by the radial distribution function in Fig. Also, quantum effects are evident when the structural properties of the liquid are investigated, this can be seen from the difference in the height of the peaks when the radial distribution functions of the classical and quantum systems are compared, as seen in Fig. 4.3), it can be noted that as the temperature is increased, the graph representing the phonostat algorithm representing the Wigner approximation converges to that of the classical simulations.

Table 4.2: Percentage difference between the comparison of the pressure obtained from classical MD simulations using cartesian coordinates (P ressure CartM D ) and classical MD simulation using the phonostat algorithm (P ressure ClassP hono ) versus tempera

Conclusion

For example, the precise definition of the effective interaction potential can be achieved by carefully and controlled variation of relevant suspension properties such as temperature and salt concentration. However, for general systems with more complex potentials, it was necessary to perform a harmonic approximation of the potential, resulting in a position-dependent harmonically approximated (PDHA) STP propagator. Also, the running times for the NH simulations were about 10% longer than the CTNH thermostat, which could be attributed to the local harmonic approximation of the potential energy at each time step.

It is also noted that thermostating any degree of freedom results in the loss of the dynamic information of the system. Our phonostat algorithm allows the nuclear quantum effects of a system to be investigated, as one can use the Wigner approach to quantum mechanics to obtain the 'effective' temperatures of the modes. A future perspective of this methodology would be to investigate the phase diagram of the Gaussian core model.

Deriving the Liouville operator for the Nosé-Hoover chain dynamics

Using the Wigner distribution function of the density matrix, one can obtain the initial quantum states for an ensemble of oscillators in the canonical ensemble at a temperature T . Let ⇢ denote the normalized density matrix, ⌦ the unnormalized density matrix and the temperature parameter β =kB1T. So⇢ˆcan be expressed as. This is possible if only even powers of the expansion vane−i2~−@·B·−!@ are taken into account. Therefore, the Wigner transform of the Bloch equation can be written as. B.0.12) This is useful in calculating quantum mechanical corrections to classical statistical mechanics.

To derive the Wigner transform of the density matrix for the general case of N independent harmonic oscillators, let us consider the Hamiltonian of a system of N independent harmonic oscillators. This equation must hold for all q and p, because terms in the bracket are independent of q and p, therefore they must vanish independently. Harmonic dynamics of proteins: Normal modes and fluctuations harmonic dynamics of proteins: Normal modes and fluctuations in bovine pancreatic trypsin inhibitor.

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