Introduction to Statistical Mechanics

aqua2001

我看这个写得比较有水平。
Biological Physics, like any other branch of condensed matter physics, deals with objects that are macroscopic, i.e. large compared to atomic dimensions and therefore consisting of very many atoms or molecules. As an example, a protein molecule with ~ 150 repeating units has more than 1500 atoms (not including the water molecules that surround the protein and interact with it) and of the order of (Avogadro's number)2 different configurational states it can adopt. Therefore, although the laws of quantum mechanics describing the dynamical behavior of atoms are well established, it is not at all practical to deduce the properties of macroscopic systems from a detailed knowledge of the interactions of each of the atoms. Instead, we need to introduce the concepts of statistical mechanics which allow us to predict the average behavior of the system based on a few simple probabilistic arguments.

As an example, and to illustrate the tools of statistical mechanics, we will start with a very simple system in which there are 4 identical molecules confined in some region of space. These molecules are in constant thermal motion and collide with each other and with the walls within which they are confined. The system is isolated from the surroundings so that the total energy of the molecules and the number of molecules remains a constant.

If we could watch the molecules as a function of time, we will find that sometimes all 4 molecules are on one side of the room, sometimes there are 3 on one side and 1 on the other side and sometimes 2 on each side. Let's calculate the probability of each of the ways in which the molecules can be distributed in space. Each of the possible configurations and the number of ways that configuration is achieved are listed below:
The number of different ways of arranging N molecules with n on one side and (N-n) on the other side is given by N!/[n!(N-n)!], where ! represents the factorial function. The total number of possible ways of arranging the molecules is 16 = 2N.

We will now apply a fundamental postulate of statistical mechanics which states that an isolated system which can be in any one of a number of accessible states (=16 in this example) is equally likely to be in any one of these states at equilibrium.

Therefore, the probability that the molecules are distributed in any one of these 16 possible ways is simply 1/16. But there are 4 ways in which the molecules can be arranged so that 3 are on the left side and 1 on the right side, and therefore, the probability of finding that configuration is 4/16. The most probable configuration is the one in which half the molecules are on one side and half on the other, i.e. the molecules are uniformly distributed over the space.

An important result from statistical mechanics is that as the number of molecules increases, the most probable distribution dominates the properties of the system.

For example, if we increase the number of molecules in the above system to N = 10, the total number of ways in which the molecules can be arranged in the two halves becomes 2N = 1024. The most probable distribution is still half on one side and half on the other. Extreme fluctuations such as all the molecules on one side are only found with a negligible probability of 1/1024.

Of course, there will always be statistical fluctuations about the most probable distribution. The size of the average fluctuation Dn is of the order N1/2. Therefore, the relative size of the fluctuation decreases as N increases.

Therefore, for large number of molecules (N »Avogadro's number), statistical fluctuations about the most probable distribution are either very small or exceedingly rare. In either case, they don't matter, and the macroscopic properties of the system are accurately described by the most probable distribution.

What about the distribution of total energy among the molecules? Clearly, the most probable distribution corresponds to the situation in which all the molecules have the same average energy. Since the average energy of a molecule is proportional to the absolute temperature, this implies that at equilibrium, the temperature is uniform over the entire region. We knew from prior experience that the temperature in a room is uniform under equilibrium conditions; now we have an explanation based on the most likely distribution of energy.

You may want to read Statistical Physics by F. Reif, Berkeley Physics Course, vol. 5 (first few chapters) for a nice and easy-to-read introduction to the concepts of statistical mechanics.

aqua2001

Mechanical and Thermal Interactions

Two or more macroscopic systems that are not isolated can interact and exchange energy. One type of interaction is mechanical interaction in which one system does macroscopic work on another system. For example, a swinging pendulum can be made to drive a nail in the wall. In this type of interaction, all the molecules of the system move in a concerted fashion in one direction. Another type of interaction is thermal interaction, in which energy is exchanged without any macroscopic work. Here energy is transferred from one system to another on an atomic scale, i.e. by collisions of the individual atoms. The energy thus transferred is called heat.

If two interacting systems are not in thermal equilibrium initially, i.e. they have different temperatures and hence different average energies per molecule (or different average energies per degree of freedom for non-identical molecules), they will exchange energy until all molecules have the same average energy (» kBT) per degree of freedom. From conservation of energy, the total energy lost by the first system is equal to the total energy gained by the second system. Therefore, the heat absorbed by a system Q increases the total energy E of that system. Q = DE is a statement of the first law of thermodynamics in the absence of mechanical work. If we include both mechanical and thermal interactions, the first law states that the change in the energy of the system is the sum of the heat absorbed by the system and the mechanical work W done on the system or Q + W = DE.

aqua2001

The Boltzmann Distribution

Our previous discussion shows that all molecules under equilibrium conditions have on average equal energy. However, at any given instant in time, it is highly unlikely that all the molecules have, in fact, the same energy. (This situation would be analogous to the probability of finding all the molecules on the same side of the room). The molecules are in constant collision with each other and constantly exchanging energies. So, what is the most likely distribution of energy at any instant in time, given the constraints that the total energy and the total number of molecules is unchanged?

The result is called the Boltzmann distribution, which we will not derive. (You can read the details of the derivation in Physical Chemistry by P. W. Atkins). Here, we will illustrate the result by another simple example. Assume that there are 3 molecules confined in space and that share a total energy E = 3e. The molecules can be in one of many discrete energy levels, uniformly spaced and separated by energy e, as shown below. (You may recall from your introductory quantum mechanics course that the energy states allowed to a particle confined in space are discrete. Discrete energy levels are true for both electrons confined in an atom or gas molecules confined in a room, so are simple illustration is valid for both situations).

There is only one way in which to arrange the molecules in configuration (a), 3 ways in which they can be arranged in configuration (b), and 6 ways in configuration (c). Therefore, the total number of accessible states of the system equals 10. According to our fundamental postulate, each one of these states is equally likely. Therefore, as the molecules collide and exchange energy, the system is likely to be found with equal probability in any one of these 10 possibilities, at a given instant in time.

Now, let's count the average populations of the various energy levels. If we take snapshots of the energy distribution as a function of time, and ask, how often is any one energy level occupied, we get a number that is proportional to 12 for the lowest energy level (there are 6 ways in which the lowest level is occupied by one molecule and 3 ways in which that level is occupied by two molecules). Similarly, we have 9 for the next higher energy level, 6 for the next one and 3 for the next one.

Therefore, the lowest energy level is occupied with the highest probability and this probability decreases as we go to higher energy levels. This result is universally true, for any number of molecules and any amount of total energy in the system.

When the number of molecules is very large, the relative populations of the various energy levels is dominated by the most probable distribution of energy, which approaches the Boltzmann distribution.

Here pi is the probability that the energy level i with energy eiis occupied, and Z is the normalization constant so that the sum of all the probabilities add up to 1.

The relative populations of the energy levels above the lowest level decrease exponentially as the energy increases.

In our previous example, if we choose e »kBT, and write down the Boltzmann factor for the lowest energy level as 1, the Boltzmann factors for energy levels 2e higher is only ~ 0.14, for 5e is 0.01 and for 10e is 4.5x10-5.

At room temperature, the accessible energies are only a few kBT.

aqua2001

Energy and Free Energy

Let's apply the Boltzmann distribution to a real biological situation. For example, you learned on the first day that proteins are linear polymers of several repeating units. The polymer has to fold up into a specific conformation in order for the protein to be biologically active. This conformation of the protein is usually referred to as the native state (N). The polymer can also be in the unfolded state (U) which would look somewhat like a strand of cooked spaghetti with no specific shape. If we assume that each repeating unit can adopt one of 3 possible orientations, and there are 100 units in this protein, the total number of possible configurations is 3100 (~ 5x1047) out of which only a few correspond to the native state. Clearly, if the energy of all the configurations were the same then the protein is equally likely to be found in any one of the 3100 possible configurations. We will assume, for the sake of simplicity, that the native state corresponds to only one of these configurations. The probability of finding the protein in the native state would be ~ 1/3100. In other words, the probability that the protein folds into a biologically active form would be negligible.

If Mother Nature wants to design a protein such that an appreciable fraction fold into the correct native state, she has to given the native state an energy bias and increase its Boltzmann factor. How much of an energy bias is needed? In the table below we calculate the ratio of PN (the probability that the protein is in the native state) and PU (the probability that the protein is in the unfolded state) as a function of the energy bias DE (=EN- EU), assuming that the ratio of possible configurations in the N and U state is 1/3100.


The exponential term is from the difference in energy between the N and U states and which favors N as its energy is decreased. The factor 1/3100, as discussed earlier, is from the difference in the number of configurations accessible in each state, and strongly favors U. Even with an energy bias of -100 kBT, N is formed with a mere probability of ~ 10-5. (One can calculate what DE should be in order to form N at least half the time by setting PN / PU = 1, or exp(-DE/kBT)x(1/3100) = 1, which gives DE»  -100kBT).

The above example shows that when applying the Boltzmann distribution to states that have not only different energies but also different number of ways in which the states can be formed, one has to take into account the ratio of the number of configurations in each state.

Therefore, more generally, the Boltzmann probability of finding a molecule in a given state is

where W is the number of possible configurations corresponding to that state and G is called the free energy of the state. In this formulation of the Boltzmann distribution, it's the free energy (which includes contributions from both the energy and number of configurations) that determines the relative population of that state. (You can easily check that G »E - kBT ln(W), and some of you may recognize that kBln(W) also defines the entropy S of that state).

The free energy function G depends on the temperature. At low temperatures the state with the lower energy has a lower free energy (and is favored at equilibrium) whereas at high temperatures the state with the larger number of configurations has the lower free energy.

aqua2001

The relative population of two states A and B of a system (for example the native and unfolded states of a protein) are given by the ratio of their Boltzmann probabilities:

where [A] and are the concentrations of the two different states of the system, and K is called the equilibrium constant for the reaction
This is an example of a unimolecular reaction in which one molecule (or macromolecule) can be in one of two possible states.

More often, the reactions in biology (and chemistry) consist of two or more molecules interacting (small molecules bind to proteins and DNA, proteins bind to DNA, two or more protein molecules can associate, two strands of the double-stranded DNA can dissociate). These are examples of bimolecular reactions of the type in which a molecule A combines with molecule B to form a complex AB. The equilibrium constant is defined as

where Ka is also referred to as the association constant or affinity constant. Note that for a bimolecular reaction, the equilibrium constant has units of [concentration]^-1.

aqua2001

有的符号打不好。上标总排不了。所以有的地方本来是10的若干次方，就把上标和底数写在一起了。大概看的时候能分辨出来吧。

pengxiangda

嗯，是有水平

zqs2008

很好 谢谢！！！！！！