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##### Module Notes

Faculty Member (Members):

Postgraduate, Spring Semester

*Module Type:*Specialization Courses

*Teaching Language:*English/Greek

*Course Code:*GCHM_C731

*ECTS Credits:*8

*Module Availability on Erasmus Students:*No

##### Module Details

The scope of the course is:

- to teach graduate students the key principles of equilibrium statistical mechanics (theory of statistical ensembles), and
- to introduce them to the techniques of the computer simulation of matter (molecular dynamics, Brownian dynamics, Monte Carlo, dissipative particle dynamics)

At the end of the course, the student will be able to:

- design molecular simulation algorithms (e.g., Molecular Dyanimcs and Monte Carlo) for simple fluids
- given a classical trajectory from such a simulation, to compute values for the thermodynamic and structural properties of the system
- use fundamental theorems of classical statistical mechanics to compute time autocrrelation and cross-correlation functions and extract from them important transport properties (e.g., viscosity and diffusivity)

- Introduction to Statistical Mechanics. Statistical Mechanics and axiomatic foundation of Thermodynamics.
- Equilibrium Statistical Ensembles.
- Dynamical trajectory in phase space. Probability density and the time-reversible Liouville equation. Ensemble averages and time averages. Ergodic hypothesis.
- Microscopic and macroscopic stress tensors, the virial theorem
- From Newton's to Hamilton's equations of motion. Derivation of the microscopic dynamical equations in the NVE ensemble
- Derivation of the microscopic dynamical equations in the NVT ensemble
- Derivation of the microscopic dynamical equations in the NPT ensemble
- Statistical ensembles for simulating systems under uniaxial extension or under flow. Derivation of the relevant microscopic dynamical equations
- The chemical potential as an ensemble average. Widom test particle insertion. Simulations of phase equilibria
- Statistical mechanics of sorption. The BET and the Langmuir isotherms
- Fluctuations and their role in determining thermodynamic properties
- Statistical mechanics of thermo-elasticity. Generalized ensembles for simulating the elastic properties of materials
- Introduction to molecular simulations. Molecular models, potential energy function, periodic boundary conditions, minimum image convention
- Molecular dynamics. Algorithms for integrating the microscopic equations of motion in the absence and presence of holonomic constraints
- The trajectory file and its post-processing. Radial pair correlation functions, time auto-correlation and cross-correlation functions
- From the trajectory file to the macroscopically observed physical properties of a system. How to get the XRD pattern. Connection with neutron spin echo experiments
- Monte Carlo sampling. Monte Carlo simulations in the canonical, isothermal-isobaric, grand canonical and semi-grand canonical ensembles
- Applications. Getting experienced with the molecular simulation of physical systems and predicting their thermodynamic, dynamic, structural, conformational, transport, interfacial, and other properties

- D.N. Theodorou and V.G. Mavrantzas,
*Multiscale modelling of Polymers*(Oxford Univ. Press, Oxford, 2019), in preparation. - D.A. McQuarrie,
*Statistical Mechanics*(Harper and Row: New York, 1976). - M.P. Allen and D.J. Tildesley,
*Computer Simulation of liquids*(Oxford Science Publications, Oxford, 1997).

Weekly or bi-weekly homework sets, semester project