The Second Law: A Statistical Perspective

A deeper understanding of entropy can be first understood in terms of a tautology; at equilibrium, a system is most likely to be found in a state in which it is statistically more likely to be found. As an illustration, imagine the 9 ’gas’ particles in the 10 × 10 2D box in Figure 1.6.

How many ways are there to organise 9 particles into 100 positions? The first particle can occupy any of the 100 positions, the second can occupy any of the remaining 99, and so on. Because the particles are indistinguishable, the ordering is irrelevant, and the total number of possible states is a 9-combination of the set of 100 positions, given by the binomial coefficient

Z = C(100, 9) ≈ 1.90 × 10¹², known as the partition function. All possible configurations of these particles are one of these states. In Figure 1.6A, the particles all occupy a small region of the box, corresponding to a perfectly ordered macrostate. There is only one configuration, or microstate, of these 9 particles corresponding to this macrostate, so we say the multiplicity, Ω, of the most ordered macrostate, is one. Whilst it is possible to swap any one of the particles for any other, as we are treating them as indistinguishable, we only consider only distinct states.

Now, we can imagine the same number of particles, occupying the same

’volume’, but in which the system is slightly less ordered, as in Figure 1.6B.

In this state, the coordinates of one of the particles is unknown; it can be in any of the 91 positions other than the top right hand corner, and the ’hole’

in the top right hand corner can be in any of the 9 positions. Thus, there are Ω = _{91 × 9} = 819 configurations, or microstates, corresponding to this slightly less ordered ’macrostate’. Clearly, all positions being equally likely, this state dominates the perfectly ordered macrostate.We can then consider an even less ordered system, in which two of the particles are disordered.

Here there are 91 × 98/2 configurations of the two particles, and 9 × 8/2 configuration of the ’hole’. Hence, there are Ω = 160, 524 states with 2 disordered particles. Again, this state is statistically dominant compared to the more ordered states. The multiplicity of increasingly disordered

’macrostates’, where the number of particles in the 3 × 3 box is specified, is shown in Table 1.2, as calculated using equation 1.45.

Ωi= ^91!

(91 − i)!i! × 9!

(9 − i)!i! =C(91, 91 − i)^×C(9, 9 − i) (1.45) It can be seen that over 99.7% of the states have four or fewer particles constrained, as calculated by dividing the sum of the multiplicities of these states by the partition function, Z. In this model system, the most disordered states are the dominant ones, so that as long as there is no a priori reason to suspect that certain microstates are more likely than others, disorder is the order of the day. Order could be defined as a different configurations, such as positioned like a chequers board, or as a 9 × 9 block not in the top right, but anywhere on the grid, or in any shape in which all particles are adjacent. The conclusions, however, are the same; the majority of configurations are non-descript. In fact, if we were to use a finer grid than 10 × 10, the dominance of

Table 1.2: The multiplicity of macrostates corresponding to i and only i particles in a pre-specified 3 × 3 box within a 10 × 10 grid.

i Ω Ω cumulative

9 1 1

8 819 820

7 147,420 148240

6 10,204,740 10,352,980 5 336,756,420 347,109,400 4 5,859,561,708 6,206,671,108 3 55,991,367,432 62,198,038,540 2 291,383,646,840 353,581,685,380 1 764,882,072,955 1,118,463,758,335 0 783,768,050,065 1,902,231,808,400=Z

the disordered configuration increases. It we divide space up so finely that, for all intent and purpose, it is continuous, then the most disordered state is so probable that the probability of finding the system in any other macro-state is vanishingly small. Further, if a system were to begin in an ordered state, and were subject to perturbations, it would spontaneously move to-wards a more disordered state, and the chance of that system spontaneously reverting back to an ordered state is very small indeed. The multiplicity of the macrostate is a precise measure of this disorder. Expressed in a more convenient logarithmic scale, the statistical definition of entropy, S, is given as

S=kBlnΩ (1.46)

The second law of thermodynamics can then be stated as ’A spontaneous process at fixed volume with a fixed number of particles has a positive change in entropy’.

For real systems, the microstate of a system is not given by its position in configurational space, but its position in phase space; each particle possesses not just a position, but also a velocity and its corresponding kinetic energy.

Similar arguments can be used for the distribution of energy as were used for spatial distribution, only the constraint now isn’t that the particles are located within the grid, but that the total energy is constant. To illustrate

this, imagine that 10 units of energy are distributed between our 9 particles.

In this case, because all the particles have different locations, they can be distinguished from one another. It could be that one single particle has all 10 units of energy, and the remaining 8 particles have none. As the particle with all the energy could be any particle, the multiplicity of this energy distribution is 9. Another possibility is that one particle has 9 units of energy, and another has 1. The multiplicity of this energy distribution is 9 × 8 = 72. Similar calculations can be made as above, and similar conclusions reached. The partition function can be calculated, and without a priori expectation about the distribution of energy, the probability of finding a system with a certain distribution of energy is calculated as the multiplicity of the distribution divided by the partition function. Just as localised matter dissipates to disorder, localised energy also dissipates to disorder. Just as when the spacing of grid points is reduced to almost continuous, the most disordered spatial distribution dominates all others, when the spacing of energy levels is reduced to a pseudo-continuum, the most disordered distribution of energy dominates. This disordered distribution of energies is the Boltzmann distribution, given by equation 1.47.

n_i The denominator is the partition function, and the numerator the mul-tiplicity. A derivation of this equation is given in Atkins and De Paula (2006), along with a demonstration of the equivalence of the classical thermodynamic definition of entropy, equation 1.44, and the statistical definition of entropy, equation 1.46. The Boltzmann distribution can be thought of as either a probability distribution when considering a single particle, or an actual energy distribution considering a quasi-infinite number of particles. It is impossible to enumerate all states in most cases, and in all cases for biological macromolecules. Thus the partition function cannot be calculated. However, the probability of finding a particle in one state relative to another can be found by the ratio of their multiplicities.