A Simple Method to calculate the number of microstates in a system with N classical particles and total energy E.

Jorge Rigol Ph.D.

A simple method is introduced to obtain the number of microstates in a system with N classical (“distinguishable”) particles and total energy E.

Assume we have a system with N classical (“distinguishable”) particles and total energy E. When we say that the particles are distinguishable, we mean that any permutation of two particles in different states must be considered as a distinct state of the whole system. We consider that the total energy E can be divided in portions each of which is a multiple of some unit of energy that we represent by the symbol εo, Each particle can get any energy as long as this energy is proportional to εo, and the total energy is equal E = noεo , (see Fig.1). This is the classical situation with a system of Simple Harmonic Oscillators.

 

no εo=E

(no-1) εo

(no-2) εo

………..

 

2 εo

1 εo

0 εo

 

Fig.1

Energy Levels

The total energy E can be distributed in many different ways. For example, one particle can get all the energy E, and the other (N-1) particles get energy zero, or one particle can get the energy (no-1)εo, while one of the other particles gets energy εo, and  the rest get zero energy, and so on. In general, there can be a large number of combinations particularly if the number of particles N and the energy take big values.

We will call a microstate [1, 2], a state where the number of particles in each specific energy state is determined. For example, suppose we have five particles and total energy E. In this case, we will have five microstates where one particle has the total energy E, and the other four particles have energy zero. Each microstate will be distinguishable from the others, because we consider that the particles are distinguishable. We are interested in finding the number of microstates associated with each possible value of the energy for a given number of particles.

The normal procedure to find the number of microstates is to count the number of possible combinations of N particles in (no+1) states of energy [3,4]. This method is particularly simple when the numbers N and no are small, but the number of combinations increases exponentially when the numbers N and no get bigger. In an effort to simplify the procedure of counting the number of microstates when the numbers N and no get bigger I have found an amazing regularity.

The best way to explain this regularity is creating a matrix (see the table 1) with number of particles vs the energy of the state. The numbers in the first column correspond to the number of particles in the system starting from 2. The numbers in the upper row correspond to the possible values for energy starting from zero increasing by 1 (in units of εo) until the maximum value E, which corresponds to the total energy in the system. The number in each cell represents the number of possible microstates in which a given particle has a specific energy.

 

Energy (E)

# of Particles

0

1

…

E-i

…

E-3

E-2

E-1

E

2

1

1

1

1

1

1

1

1

1

3

…

…

…

…

…

4

3

2

1

4

…

…

…

…

…

10

6

3

1

5

…

…

…

…

…

20

10

4

1

6

…

…

…

…

…

35

15

5

1

7

…

…

…

…

…

56

21

6

1

…

…

…

…

…

…

…

…

…

1

 

N

 

 

 

…

 

 

…

 

…

 

(N-2+i)!/((N-2)!*i!)

 

…

 

(N+1)(N)(N-1)/3!

 

N*(N-1)/2!

 

(N-1)

 

1

 

Table 1

Distribution of microstates for different values of N and E.

 

The regularity I have observed is that the number in each cell is just the sum of the number in the cell above and the number of the cell to the right. For example,  starting from the right-upper side, we get 2 adding 1 + 1, we get 3 adding 2+1. For N=7, the number 56 is obtained by adding 35 plus 21, and so on. That’s it!

To see how this works in a concrete example, let’s consider that we have a system with total energy E= 9εo and the number of particles goes from 2 to 10.

 

  Energy

 

# Particles

 

 

0

 

 

1

 

 

2

 

 

3

 

 

4

 

 

5

 

 

6

 

 

7

 

 

8

 

 

9

 

 

TOTAL

2

1

1

1

1

1

1

1

1

1

1

10

3

10

9

8

7

6

5

4

3

2

1

55

4

55

45

36

28

21

15

10

6

3

1

220

5

220

165

120

84

56

35

20

10

4

1

715

6

715

495

330

210

126

70

35

15

5

1

2002

7

2002

1287

792

462

252

126

56

21

6

1

5005

8

5005

3003

1716

924

462

210

84

28

7

1

11440

9

11440

6435

3432

1716

792

330

120

36

8

1

24310

10

24310

12870

6435

3003

1287

495

165

45

9

1

48620

 

Table 2

Distribution of microstates for different values of N and E=9 εo.

Each cell contains the number of microstates for a particular value of the energy for a given number of particles. The last column shows the total number of microstates for a given number of particles. We can see in the table, how fast the number of microstates decreases when the energy increases.

 

If we have a different number of particles (N), or different total energy (E), the procedure to follow to complete the table is equally simple: we put ones in the last column that corresponds to the maximum energy and ones in the first row that corresponds to two particles. After that, to get the number for each specific cell, we just add the number in the cell above with the number on the cell to the right.

Practically in every Statistics Physics book we can find examples of how to calculate the number of microstates for specific values of the energy and the number of particles. For example, the value 2002 for N=6 coincides with the value reported in [5].

Each cell shows ALL possible microstates available for ONE particle and some specific energy. I want to make it clear that using this method, we can find the number of microstates accessible for the particle for a given energy, and the total number of microstates accessible, but we can’t determine in which microstate the other particles are. However, usually, the last is not the most interesting question to answer. 

From a statistical point of view, the bigger the number of microstates, the higher will be the probability of finding the particle in that particular state. So, we can expect that the relative probability of finding a particle in two different states will be proportional to the relative number of microstates between these two states. Following this idea, we can find the relative probability of finding a particle in states with different energy.

            To compare the relative probability of finding a particle in different energy states, we normalize the number of microstates for a given energy, to the number of microstates obtained for energy zero. Looking at the table we can see that for a specific number of particles this relative probability decreases very fast when the energy increases.

In the following graph we compare the relative probability of finding a particle in different energy states for different values of N. For each specific number of particles the different values are normalized to the number of microstates obtained for the value of energy equal zero.

image010.gif

Fig.2

Probability Distribution for different values of N

From the figure, it is evident that the plots are getting closer and closer to an exponential function as the number of particles increases.

            Using the results obtained in Table 1, the number of microstates in each cell can be calculated by using the explicit formula:

     

where N is the number of particles and “i” takes the integer values from 0 to no.

It is well known from the literature [6,7], that the total number of microstates for a specific of energy no and the number of particles equal N should be equal:

So, if our method is correct, the sum of all microstates obtained for given values of N and no, should be equal to that number. Or, in another words, the following equality should be true:

Using a simple program in the graphing calculator, I have checked that both formulas give actually the same result. 

ACKNOWLEDGMENTS

I would like to thank Dr. M. Rigol  for very useful and encouraging discussions.

 

[1] F. Pierce; Microscopic Thermodynamics (International Textbook Company, 1968), p. 127

[2] M. Glazer and J. Wark; Statistical Mechanics (Oxford University Press, 2001), p.3

[3] Ref.1, p.130

[4] Ref.2 ,p.10

[5] F. J. Blatt, Modern Physics (McGraw-Hill, 1992), p.224-226

[6] D. Yoshioka, Statistical Physics (Springer, 2007), p.25

[7] M.D.Sturge, Statistical and Thermal Physics (A.K.Peters Ltd, 2003) p.43