Lecture 2 Molecular dynamics simulates a system by numerically following the path of all particles in phase space as a function of time the time T must be long enough to allow the system to explore all accessible regions of phase space the time average of a quantity A is calculated from

Monte Carlo no dynamics but random motion in configuration space due to random but uncorrelated forces generate configurations or states with a weight proportional to the canonical or grand canonical probability density actual steps of calculation depend on the model

Review of Probability and Statistics

Introduction Probability and statistics are the foundations of both statistical mechanics and the kinetic theory of gases what does the notion of probability mean? Classical notion: we assign, a priori, equal probabilities to all possible outcomes of an event Statistical notion: we measure the relative frequency of an event and call this the probability

Classical Probability Count the number W of outcomes and assign them equal probabilities p i = 1/W for example: a coin toss each “throw” is a trial with W=2 outcomes p H = p T = 1/2 for N consecutive trials, a particular sequence of heads and tails constitutes an event HTTHHHTT... there are 2 N possible outcomes and the probability of each “event” is p i = 1/2 N

Classical Probability We cannot predict which sequence (event) will occur in a given trial hence we need a statistical description of the system => a description in terms of probabilities instead of focusing on a particular system or sequence, we can think of an assembly of systems called an ensemble repeat the N coin flips a large number (M) of times if event ‘i’ occurs m i times in these M members of the ensemble, then the fraction m i /M is the probability of the event ‘i’

Probability of a head (H) is the number of coins n H with a H divided by the total number M in the ensemble

Classical Probability in statistical mechanics we use this idea by assuming that all accessible quantum states of a system are equally likely basic idea is that if we wait long enough, the system will eventually flow through all of the microscopic states consistent with any constraints imposed on the system measurements must be treated statistically the microcanonical ensemble corresponds to an isolated system with fixed total energy E however this is not the most convenient approach

Statistical Probability Experimental method of assigning probabilities to events by measuring the relative frequency of occurrence if event ‘i’ occurs m i times in M trials, then

Independent Events If events are independent, then the probability that both occur p i,j = p i p j e.g coin toss with 2 trials => 4 outcomes p H,H =p T,T =p H,T =p T,H = (1/2)(1/2)=1/4 but probability of getting one head and one tail in 2 trials = 1/4 + 1/4 = 1/2 (order unimportant!) probability of 2 heads and 2 tails (independent of order) in 4 tosses is

Random Walks Consider a walker confined to one dimension starting at the point x=0 the probability of making a step to the right is p and to the left is q=1-p ( p+q=1) each step is independent of the preceding step let the displacement at step i be denoted as s i where s i = ±a each step is of the same magnitude where is the walker after N steps? a

Random Walk Net displacement

Averages The average of a sum of independent random variables is equal to the sum of the averages

Averages The average of the product of two statistically independent random variables is equal to the product of the averages

Dispersion or Variance Walker does not get very far from its mean value if N>>1 !

Probability Distribution What is the probability P(x,N) that walker ends up at point x in N steps? Total number of steps N= n R + n L probability of n R steps to right is p nR probability of n L steps to left is q nL number of ways = N!/n R !n L ! but x = (n R - n L )a hence n R = (N+ x/a)/2 n L = (N - x/a)/2set a=1

Set a=1

N=20 N=40

-N<x<N Define r=x/N where -1<r<1

Show

For large N, p(x,N) approaches a continuous distribution