MONTE CARLO SIMULATIONS Current physics research 1) Theoretica 2) Experimenta 3) Computationa Monte Caro (MC) Method (1953) used to study 1) Discrete spin systems 2) Fuids 3) Poymers, membranes, soft matter 4) Disordered materias 5) Lattice gauge theories 6) Compex systems, networks, etc Samping Thermodynamic variabe {s} A = A exp[ βh] {s} exp[ βh]
where H is the Hamitonian for the system, β = 1/kT is the inverse temperature, and {s} denotes the configurations. We consider the Ising mode H = J s i s j h ij i s i s i = ±1 Ising variabe, i, j on attice, J exchange, h magnetic fied. Average vaues {s} s i = s = s i exp[ βh] {s} exp[ βh] {s} s i s j = s is j exp[ βh] {s} exp[ βh] {s} H = H exp[ βh] {s} exp[ βh] Ising mode on attice with N sites, 2 N configurations. NB! Ony statisticay independent for T, β 0. W = 2 N impies S = Nk og 2 Direct evauation ony feasibe for N 40
Standard MC (non importance samping) 1) Pick set of random configurations {s} 2) Compute {s} A({s}) exp[ βh] A = {s} exp[ βh] Standard techniques for evauating integras. Does not work here because of rapid variation of exp[ βh] Probabiity distribution Ony restricted region of configuration space {s} contribute to A. We must sampe from important regions in configuration space.
Importance MC samping Genera Markov chain of configurations: Consider a set of configurations {} abeed by this coud for exampe be a set of spin configurations {s} 1, {s} 2, {s} 3, {s} n 1, {s} n = {{s}} A Markov chain is defined by specifying the transition probabiity w m to go from configuration m to configuration. There are no memory effects. Simuation starts in state 0 at time t = 0, we generate a Markov chain of subsequent states with time abe t = 0, 1, 2, 3,, i.e., 0, 1, 2,. We assign the probabiity P (t) to the state at time t. At t = 0 system is definitey in state 0, i.e., P (0) = δ,0
At ater times we assign a probabiity P (t). At ong times we desire P (t) to approach the equiibrium distribution P eq, im t P (t) = P eq The average vaue of A is A = P eq A To obtain a thermodynamic average we require P eq = e βe m e βe where E is the energy of the state. We must choose transition probabiities w m right to obtain this equiibrium distribution. The evoution of P (t) is determined by the Master equation P (t+1) P (t) = [P m (t)w m P (t)w m ] m
First term describes transitions into state, second term describes transitions out of state. Note that ony terms with m contribute. We can aso define w to be the probabiity that the system stays in state, i.e., w = 1 m w m or m w m = 1 Combining, we can write the Master equation in the form P (t + 1) = m P m (t)w m Ceary, the Master equation preserves the normaization of probabiities P (t + 1) = P (t) = 1 since,m P m (t)w m =,m P (t)w m. For the required stationary distribution we obtain from the Master equation the condition m (P eq w m P eq m w m ) = 0
or since m w m = 1 P eq = m P eq m w m In practice stationarity is accompished by demanding the detaied baance condition P eq w m = Pm eq w m or, inserting the Botzmann distribution w m w m = e β(e m E ) A common way of impementing a Monte Caro move is to choose a tria state m as the possibe state for the system at time t+1. The probabiity that the tria state is m if the state at time t was, is given by a proposa matrix U m. This satisfes the condition m U m = 1, and is usuay chosen to be symmetric. State m is then accepted as the state at t + 1 with some probabiity a m (expained beow), i.e. w m = U m a m Otherwise the state at t + 1 is state.
For exampe, in an Ising probem, state m is frequenty chosen to be a state in which one of the spins (chosen at random) in state has been reversed. In this case U m = 1/N, where N is the number of sites, if and m differ by a singe spin fip, and 0 otherwise. One generates a random number, r, with a uniform distribution between 0 and 1, and if r < m the move is accepted, i.e. the state at time t+1 is m, and otherwise the move is rejected, i.e. the state at t+1 is, the same as at time t. After testing N spins we say that a singe Monte Caro sweep of the attice has been performed. A sweep is the natura unit in which to describe the ength of a simuation. It is aso possibe to pass sequentiay through the attice (i.e. to test spins 1, 2, 3, 4,, N in that order) rather than to go through the attice in a random sequence. This saves the generation of a random number. Going back to the genera discussion, the energy difference is E = E m E and the detaied baance condition for a is ceary a m = w m = e β E a m w m
(for a symmetric proposa matrix). satisfied by This is a m = F ( e β E) where F is any function which satisfies 0 F (x) 1 (since probabiities cannot be negative or greater than unity) and F (x) F (1/x) = x for a x Two possibe choices are 1. The Metropois agorithm, F (x) = min(x, 1) In this approach one aways accepts the move if it gains energy but ony accepts it with probabiity exp( β E) if it costs energy, i.e. if E > 0. 2. F (x) = x 1 + x
which corresponds to an acceptance probabiity of 1 e β E + 1 irrespective of the sign of E. For an Ising mode, where each spin can ony be in one of two states, this is an exampe of the heatbath method, where, after the move, the probabiity of the variabe being atered is independent of its vaue before the move and just corresponds to a oca therma equiibrium for that variabe in its instantaneous environment. The proof that P (t) actuay converges to P eq wi not be given here.
ON A COMPUTER 1) set up attice sites i, define spins s i define Hamitonian H set counter n = 1, choose n 0, nmax 2) fip spin at random or sequentiay cacuate r = e E/kT generate a random number 0 < z < 1 if r > z accept fip, if r < z reject fip 3) cacuate variabes A n store for each step n > n 0 4) return to 2) 5) cacuate average A = 1 n n>n 0 A n Hans Fogedby