PHYS3410 BIOPHYSICS II
Lecture Notes

Section I: Thermodynamics of Biological Systems

Lecture 5 : Statistical  Interpretation of Entropy

So far the concept of entropy as a state function is entirely macroscopic. The validity of Second Law stems from the reality of irreversible processes.

In stark contrast to these processes, which we see all around us, the laws of both classical and quantum mechanics are time symmetric: a system that can evolve from state A to a state B, can also evolve from state B to state A. For example, the spontaneous flow of gas molecules from a higher density region to a lower density region and its reverse (which violates the Second Law), are both in accord with the laws of mechanics.

Yet all irreversible macroscopic processes are consequences of molecular movement, collisions and transfer of energy. How can irreversible macroscopic processes emerge from the reversible motion of molecules?

Ludwig Boltzmann (1844 - 1906) attempted to reconcile the macroscopic and microscopic approaches and derive mechanical interpretation of entropy. Boltzmann was deeply influenced by the Darwin’s idea of evolution.

Entropy, Disorder, Probability

Spontaneous processes in nature are often associated with increase of disorder.
This tendency toward disorder and homogeneity can be related to time's arrow.

What do we mean by order and disorder?
Descriptions of the states of a system can be divided into two classes:

Microscopic descriptions: Describe the system in highest degree of detail. (e.g. positions and velocities of all the atoms in the system). This is usually impossible for us to do.

Macroscopic descriptions: Overall state of the system: averaging over large number of atoms or molecules (e.g. pressure) or identifying the internal energy of ideal gas with the kinetic energy of the molecules. These are measurements and calculations we can make.

To gain deeper understanding of a particular system, we need to relate our macroscopic measurements to theoretical descriptions of the microscopic entities.

How do microstates relate to any particular macrostate?

Often an observable macrostate can arise from very different arrangement of microstates.

If many macrostates are possible and these can be constructed from different configuration of microstates, why do we observe some macrostates and not others?

All microstates corresponding to macrostates of a given energy are equally probable.

Therefore, the greater the number of microstates which lead to a particular macrostate, the greater the probability of observing that macrostate.

Consider a system of eight objects:

The thermodynamic probability, W,of observing a particular macrostate is proportional to number of microstates that belong to it. For the above example it can be determined by counting the number of ways the objects can be rearranged within the sixteen squares without altering the macrostate. The most uniform or homogeneous macrostate is the most probable. This state corresponds to the least ordered state. So: the more ordered states are the least probable!

In actual thermodynamic systems the numbers of particles (and consequently microstates) are much larger (N_A = 6.02 x 10²³ particles per mole), so probability distribution is sharply peaked at the homogeneous configuration.

Boltzmann defined entropy of an observable macrostate in terms of the number of its' microstates W:

S = k_B ln W  (5.1)

k_B =  Boltzmann constant = 1.381 x 10^-23 JK^-1

Macrostates with microstates, which are not statistically uniform (microscopic order), are usually not in equilibrium and will spontaneously evolve to microscopic disorder (equilibrium).

Consider the macrostate of a box containing a gas with N₁ molecules in one half and N₂ in the other.

 (5.2)

The macrostates with large W are more probable. The irreversible increase in entropy then corresponds to evolution to states of higher probability. Equilibrium states are those for which W is a maximum (for eqn. 2.5, W reaches a maximum when N₁ = N₂).

Planck's point of view:

Define Entropy S, which always changes in the same sense in all natural processes and assumes maximum at system equilibrium. Entropy is a function of thermodynamic probability S = f (W)

We already know that S is an extensive additive function of state: S₁₊₂ = S₁ + S₂

But from probability theory: W₁₊₂ = W₁W₂!

Combining the two relations, the functional form: S = k ln W                             

This eqn. is also in accord with the Third Law: S = 0, W = 1 (only one microstate possible) at absolute zero of temperature, where U is minimum. The connection between entropy and disorder plays an important role in interpreting the thermodynamic behavior of living systems.

Exercise: In a “3 up” game, three coins are tossed simultaneously and chosen combination of heads and tails wins. If each toss is a macrostate, consider the number of microstates for different type of combinations.