Statistical physics Basic ideas of statistical physics Statistics

Statistical physics

Basic ideas of statistical physics Statistics is a branch of science which deals with the collection, classification and interpretation of numerical facts. When statistical concept are applied to physics then a new branch of science is called Physics Statistical Physics.

Trial → experiment→ tossing of coin Event → outcome of experiment Exhaustive events The total number of possible outcomes in any trial For tossing of coin exhaustive events = 2

Favourable events number if possible outcomes (events) in any trial Number of cases favourable in drawing a king from a pack of cards is 4. Mutually exclusive events no two of them can occur simultaneously. Either head up or tail up in tossing of coin. Equally likely events every event is equally preferred. Head up or tail up

Independent events if occurrence of one event is independent of other Tossing of two coin Probability The probability of an event =

If m is the number of cases in which an event occurs and n the number of cases in which an event fails, then Probability of occurrence of the event = Probability of failing of the event = The sum of these two probabilities i. e. the total probability is always one since the event may either occur or fail.

Tossing of two coins : The following combinations of Heads up (H) and Tails up(T) are possible :

Principle of equal a priori probability The principle of assuming equal probability for events which are equally likely is known as the principle of equal a priori probability. A priori really means something which exists in our mind prior to and independently of the observation we are going to make.

Distribution of 4 different Particles in two Compartments of equal sizes Particles must go in one of the compartments. Both the compartments are exactly alike. The particles are distinguishable. Let the four particles be called as a, b, c and d. The total number of particles in two compartments is 4 i. e.

The meaningful ways in which these four particles can be distributed among the two compartments is shown in table.

Macrostate The arrangement of the particles of a system without distinguishing them from one another is called macrostate of the system. In this example if 4 particles are distributed in 2 compts, then the possible macrostates (4+1) If n particles are to be distributed in 2 compts. Then the no. of macrostates is =5 = n+1

Microstate The distinct arrangement of the particles of a system is called its microstate. For example, if four distinguishable particles are distributed in two compartments, then = 24 the no. of possible microstates (16) If n particles are to be distributed in 2 compartments. The no. of microstates is = 2 n =(Compts)particles

Thermodynamic probability or frequency The numbers of microstates in a given macrostate is called thermodynamics probability or frequency of that macrostate. For distribution of 4 particles in 2 identical compartments W(4, 0) =1 W(3, 1) =4 W(2, 2) = 6 W(1, 3) = 4 W(0, 4) =1

W depends on the distinguishable or indistinguishable nature of the particles. For indistinguishable particles, W=1 macrostate Frequency probability Micro. States W Comp 1 Comp 2 (4, 0) 1 (3, 1) 1 (2, 2) 1 (1, 3) 1 (0, 4) 1

All the microstates of a system have equal a priori probability. Probability of a microstate = Probability of a macrostate = (no. of microstates in that microstate) (Probability of one miscrostate) = thermodynamic probability× prob. Of one microstate

Constraints Restrictions imposed on a system are called constraints. Example total no. particles in two compartments = 4 Only 5 macrostates (4. 0), (3, 1), (2, 2), (1, 3), (0, 4) possible The macrostates (1, 2), (4, 2), (0, 1), (0, 0) etc not possible

Accessible and inaccessible states The macrostates / microstates which are allowed under given constraints are called accessible states. The macrostates/ microstates which are not allowed under given constraints are called inaccessible states Greater the number of constraints, smaller the number of accessible microstates.

Distribution of n Particles in 2 Compartments The (n+1) macrostates are (0, n) (1, n, 1)… (n 1, n 2)…… (2, n 2), …. . (n 0), Out of these macrostates, let us consider a particular macrostate (n 1, n 2) such that n 1 + n 2 = n n particles can be arranged among themselves in n. P n = n! ways

These arrangements include meaningful as well as meaningless arrangements. Total number of ways = (no. of meaningful ways) (no. of meaningless ways) n 1 particles in comp. 1 can be arranged in = n 1 ! meaningless ways. n 2 particles in comp. 2 can be arranged in = n 2 ! meaningless ways. n 1 particles in comp. 1 and n 2 particles in comp. 2 can be arranged in = n 1 ! n 2 ! meaningless ways.

The total no. of microstates = 2 n Prob. of distribution (r, n-r)

Deviation from the state of Maximum probability The probability of the macrostate (r, n r) is When n particles are distributed in two comp. , the number of macrostates = (n+1) The macrostate (r, n r) is of maximum probability if r = n/2, provided n is even. The prob. of the most probable macrostate

Probability of macrostate is slightly deviate from most probable state by x. (x<<n) Then new macrostate will be

stirling’s formula Taylor’s theorem Do at home to get

Discussion n 103 0. 999 106 0. 607 108 1010

Thus we conclude that ass n increases the prob. of a macrostate decreases more rapidly even for small deviations w. r. t. the most probable state. n 1 > n 2 > n 3 n 2 n 1 0. 2 0. 1 0 (2 x / n) 0. 1 0. 2

Static and Dynamic systems Static systems: If the particles of a system remain at rest in a particular microstate, it is called static system. Dynamic systems: If the particles of a system are in motion and can move from one microstate to another, it is called dynamic system.

Equilibrium state of a dynamic system A dynamic system continuously changes from one microstate to another. Since all microstates of a system have equal a priori probability, therefore, the system should spend same amount of time in each of the microstate. If tobs be the time of observation in N microstates The time spent by the system in a particular macrostate Let microstate has frequency

Time spend in macrostate That is the fraction of the time spent by a dynamic system in the macrostate is equal to the probability of that state

Equilibrium state of dynamic system The macrostate having maximum probability is termed as most probable state. For a dynamic system consisting of large number of particles, the probability of deviation from the most probable state decrease very rapidly. So majority of time the system stays in the most probable state. If the system is disturbed, it again tends to go towards the most probable state because the probability of staying in the disturbed state is very small. Thus, the most probable state behaves as the equilibrium state to which the system returns again and again.

Distribution of n distinguishable particles in k compartments of unequal sizes The thermodynamic prob. for macrostate Let the comp. 1 is divided into no. of cells Particle 1 st can be placed in comp. 1 in = no. of ways Particle 2 nd can be placed in comp. 1 in = no. of ways Particle no. of ways can be placed in comp. 1 in =

particles in comp. 1 can be placed in = particles in comp. 2 can be placed in = particles in comp. k can be placed in = total no. ways in which n particles in k comparmrnts can be arranged in the cells in these compartments is given by

Thermodynamic probability for macrostate is