From Atoms to Materials: Predictive Theory and Simulations Week 4: Connecting Atomic Processes to the Macroscopic World Lecture 4.2: The Canonical Ensemble and Microscopic Definition of T Ale Strachan [email protected] School of Materials Engineering & Birck Nanotechnology Center Purdue University West Lafayette, Indiana USA Statistical Mechanics: microcanonical ensemble Equal probability postulate: N,V,E Relationship between microscopic world and thermo: S = k logΩ(E,V , N ) Statistical Mechanics: canonical ensemble E+Ebath=Etot=Constant Sys Bath Probability of system being in a microscopic ({Ri},{Pi}) state with energy E: Since E<<Etot we expand logΩbath around Etot: Maxwell – Boltzmann distribution Canonical ensemble and thermodynamics Maxwell-Boltzmann distribution: Partition function: E P(E) − kT E e Ω(N,V , E) log Z(N,V ,T ) = logΩ(N,V , E)− kT Helmholtz free energy: E Canonical ensemble: averages Consider a quantity that depends on the atomic positions and momenta: In equilibrium the average values of A is: Ensemble average When you measure the quantity A in an experiment or MD simulation: Time average Under equilibrium conditions temporal and ensemble averages are equal Various important ensembles Microcanonical (NVE) Canonical (NVT) Isobaric/isothermal (NPT) Probability distributions E E−PV − − kT kT Z(T,V , N ) = ∑e Z P (T, P, N ) = ∑ ∑e micro V micro Free energies (atomistic ↔ macroscopic thermodynamics) S = k logΩ(E,V , N ) F(T,V , N ) = −kT log Z G(T, P, N ) = −kT log Z p Canonical ensemble: equipartition of energy Average value of a quantity that appears squared in the Hamiltonian Change of variable: Equipartition of energy: Any degree of freedom that appears squared in the Hamiltonian contributes 1/2kT of energy Equipartition of energy: MD temperature 3N K = kT 2 In most cases c.m. motion is set to zero at time zero (constant of motion → it remains zero) 3N − 3 K = kT 2 Often angular momentum is zeroed (and remains zero) 3N − 6 K = kT 2 Temperature is related to average kinetic energy. Instantaneous temperature: N K(t) = eff kT(t) 2.