The Basic (Fundamental) Postulate of Statistical Mechanics

A system which has no interaction Definition: ISOLATED SYSTEM ≡ A system which has no interaction of any kind with the “outside world” This is clearly an idealization! Such a system has No Exchange of Energy with the outside world.  The laws of mechanics tell us that the total energy E of this system is conserved. E ≡ Constant So, an isolated system is one for which Total Energy is Conserved.

All microstates accessible to it Consider an isolated system. The total energy E is constant & the system is characterized by this energy. So All microstates accessible to it MUST have this energy E. For many particle systems, there are usually a HUGE number of states with the same energy. Question: What is the probability of finding the system in any one of these accessible states?

“Equilibrium” Independent of Time. Before answering this, lets define: “Equilibrium” A System in Equilibrium is one for which the Macroscopic Parameters characterizing it are Independent of Time.

Isolated System in Equilibrium: In the absence of any experimental data on some specific system properties, all we can really say about this system is that it must be in one of it’s accessible states (with that energy). If this is all we know, We can “handwave” the following: There is nothing in the laws of mechanics (classical or quantum) which would lead us to suspect that, for an ensemble of similarly prepared systems, we should find the system in some (or any one) of it’s accessible states more frequently than in any of the others.

EQUALLY LIKELY TO BE FOUND IN ANY ONE OF IT’S ACCESSIBLE STATES So,  It seems reasonable to ASSUME that the system is EQUALLY LIKELY TO BE FOUND IN ANY ONE OF IT’S ACCESSIBLE STATES In (equilibrium) statistical mechanics, we make this assumption & elevate it to the level of a POSTULATE.

THE FUNDAMENTAL (!) (basic) POSTULATE OF (equilibrium) Statistical Mcehnaics: “An isolated system in equilibrium is equally likely to be found in any one of it’s accessible (micro-) states” This is sometimes called the “Postulate of Equal à-priori Probabilities” This is the basic postulate (& the only postulate) of equilibrium statistical mechanics!!

vital importance of this in Statistical Mechanics! I want to emphasize the vital importance of this in Statistical Mechanics! I can’t stress enough that This is the Fundamental, Basic (& only) Postulate of equilibrium statistical mechanics. “An isolated system in equilibrium is equally likely to be found in any one of it’s accessible states”

à-priori Probabilities” “An isolated system in equilibrium is equally likely to be found in any one of it’s accessible microstates” This is sometimes called the “Postulate of Equal à-priori Probabilities”

ALL of Physics is an Experimental Science With this postulate, we can (& will) derive ALL of 1. Classical Thermodynamics, 2. Classical Statistical Mechanics, 3. Quantum Statistical Mechanics. It is reasonable & it doesn’t contradict any laws of classical or quantum mechanics. But, is it valid & is it true? To answer this, remember that Physics is an Experimental Science

Physics is an Experimental Science So, lets accept it & continue on. The only practical way to see if this hypothesis is valid is to develop a theory based on it & then to remember that Physics is an Experimental Science Whether the Fundamental postulate is valid or not can only be decided by Comparing the predictions of a theory based on it with experimental data. A HUGE quantity of data taken over 250+ years exists! None of this data has been found to be in disagreement with the theory based on this postulate. So, lets accept it & continue on.

Simple Examples: Example 1 Back to the example of 3 spins, an isolated system in equilibrium. Suppose that the total energy is measured as: E ≡ - μH. We’ve seen that the only 3 possible system states consistent with this energy are: (+,+,-) (+,-,+) (-,+,+)  The Fundamental Postulate says that, when the system is in equilibrium, it is equally likely (with probability = ⅓) to be in any one of these 3 states.

(+,+,-) (+,-,+) (-,+,+) E ≡ - μH. 3 spins, isolated & in equilibrium. Total energy: E ≡ - μH. 3 possible system states consistent with this energy: (+,+,-) (+,-,+) (-,+,+)  The Fundamental Postulate says it is equally likely (with probability = ⅓) to be in any one of these 3 states. Note: This probability is about the system, NOT about individual spins. Under these conditions, it is obviously NOT equally likely that an individual spin is “up” & “down”. It is twice as likely for a given spin to be “up” as “down”.

Example 2 Consider N (~ 1024) spins, each with spin = ½. Put the system in an external magnetic field H. The total energy is measured & found to be: E ≡ - μH. This is similar to the 3 spin system, but now there are a HUGE number of accessible states. The number of accessible states is equal to the number of possible ways for the energy of N spins to add up to - μH.  The Fundamental Postulate says that, when the system is in equilibrium, it is equally likely to be in any one of these HUGE numbers of states.

Example 3 E = ½(p2)/(m) + ½κx2 (1) E = Constant Classical Illustration: Consider a 1-dimensional, classical, simple harmonic oscillator mass m, with spring constant κ, position x & momentum p. Total energy: E = ½(p2)/(m) + ½κx2 (1) E is determined by the initial conditions. If the oscillator is isolated, E is conserved. How do we find the number of accessible states for this oscillator? Consider the (x,p) phase space. In that space, E = constant, so (1) is the equation of an ellipse: p x E = Constant

δE ≡ Uncertainty in the Energy. If we knew the oscillator energy E exactly, the accessible states would be the points on the ellipse. In practice, we never know the energy exactly! There is always an experimental error δE. δE ≡ Uncertainty in the Energy. We always assume: |δE| <<< |E| For the geometrical picture in the x-p plane, this means that the energy is somewhere between 2 ellipses, one corresponding to E & the other corresponding to E + δE.

δE ≡ Uncertainty in the energy. Always: |δE| <<< |E| In the x-p plane, the energy is somewhere between the 2 ellipses, one corresponding to E & the other corresponding to E + δE. See the figure: # accessible states ≡ # phase space cells between 2 ellipses ≡ (A/ho) A ≡ area between ellipses & ho ≡ qp

The Fundamental Postulate of Statistical Mechanics: In general, there are many cells in the phase space area between the ellipses (ho is “small”). So, there are A HUGE NUMBER of accessible states for the oscillator with energy between E & E + δE. That is, there are many possible values of (x,p) for a set of oscillators in an ensemble of such oscillators. The Fundamental Postulate of Statistical Mechanics:  All possible values of (x,p) with energy between E & E + δE are equally likely. Stated another way, ANY CELL in phase space between the ellipses is equally likely.

Statistical Ensembles + More on the Fundamental Postulate A typical episode from the on-line comic series “PhD Comics” by Jorge Cham There are also movies! “PhD Movie I” & “PhD Movie II”

Calculating the Most Probable We know, that Statistical Mechanics deals with systems of a large number N of particles. N is so huge, that we give up trying to keep track of individual particles. We can’t solve Schrödinger’s Eqtn in closed form for helium (4 particles), so what hope do we have of solving it for the gas molecules in this room (10f particles) ?? Statistical Mechanics handles many particles by Calculating the Most Probable Behavior of the System as a Whole rather than by being concerned with the behavior of individual particles.

In Statistical Mechanics We assume that the more ways there are to arrange the particles to give a particular distribution of energies, the more probable that distribution is. Example: 6 energy units, 3 particles to give it to Color Codes for Energy Units 6 ways 3 2 1 3 1 2 2 1 3 2 3 1 1 2 3 1 3 2 1 1 4 4 1 1 1 4 1 3 ways ≡ 1 ≡ 2 ≡ 3 ≡ 4 Most Probable Distribution

All energy distributions are Another Example The Fundamental Postulate says that All energy distributions are equally probable If E = 5 and N = 5 then 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 All possible configurations have equal probability, but the possible number of ways (weight) is different for each.

The Dominant Configuration For a large number of molecules & a large number of energy levels, there is a Dominant Configuration. In the probability distribution, the weight of the dominant configuration is much larger than the weight of the other configurations. Weight of the Dominant Configuration Weights Wi {ni}  Configurations

The Dominant Configuration If E = 5 and N = 5 then 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 W = 1 = (5!/5!) W = 20 = (5!/3!) W = 5 = (5!/4!) The difference in the W’s becomes larger as N increases! In molecular systems (N~1023) considering the most dominant configuration is certainly enough to calculate averages.

The Principle of Equal à-priori Probabilities Statistical thermodynamics is based on the fundamental postulate of Equal à-priori Probabilities. That is, all possible configurations of a given system which satisfy the given boundary conditions such as temperature, volume and number of particles, are equally likely to occur. OR The system is equally likely to be found in any one of its accessible states.

Example Consider the orientations of three unconstrained & distinguishable spin-1/2 particles. What is the probability that 2 are spin up & 1 is down at any instant?

↑↑↑, ↑↑↓, ↑↓↑, ↓↑↑, ↑↓↓, ↓↑↓, ↓↓↑, ↓↓↓ Example Consider the orientations of three unconstrained & distinguishable spin-1/2 particles. What is the probability that 2 are spin up & 1 is down at any instant? Solution Of the eight possible spin configurations for the system: ↑↑↑, ↑↑↓, ↑↓↑, ↓↑↑, ↑↓↓, ↓↑↓, ↓↓↑, ↓↓↓ The second, third, & fourth make up the subset "two up and one down". Therefore, The probability of occurrence of this particular configuration is: P = 3/8

The Coupled Many Particle Newton’s 2nd Law Equations of Motion. In principle, the problem of a many particle system is completely deterministic: If we specify the many particle wavefunction Ψ (state) of the system (or the classical phase space cell) at time t = 0, we can determine Ψ for all other times t by solving The Time-Dependent Schrödinger Equation & from Ψ(t) we can calculate all observable quantities. Or, classically, if we specify the positions & momenta of all particles at time t = 0, we can predict the future behavior of the system by solving The Coupled Many Particle Newton’s 2nd Law Equations of Motion.

ENSEMBLE. MACROSCOPIC properties.  We use Probability & Statistics. Generally, we usually don’t have such a complete specification of the system available. We need f quantum numbers, but f ≈ 1024! Actually, we usually aren’t interested in such a complete microscopic description anyway. Instead, we’re interested in predictions of MACROSCOPIC properties.  We use Probability & Statistics.  To do this we need the concept of an ENSEMBLE.

A goal of Statistical Mechanics is to Predict this Probability. A Statistical Ensemble is a LARGE number (≡ N) of identically prepared systems. In general, the systems of this ensemble will be in different states & thus will have different macroscopic properties.  We ask for the probability that a given macroscopic parameter will have a certain value. A goal of Statistical Mechanics is to Predict this Probability.

or μz = -μ, for spin “down” Example Consider the spin problem again. But, now, Let The System Have N = 3 Particles, fixed in position, each with spin = ½  Each spin is either “up” (↑, m = ½) or “down” (↓, m = -½). Each particle has a vector magnetic moment μ. The projection of μ along a “z-axis” is either: μz = μ, for spin “up” or μz = -μ, for spin “down”

 There are (2)3 = 8 Possible States!! Put this system into an External Magnetic Field H. Classical E&M tells us that a particle with magnetic moment μ in an external field H has energy: ε = - μH Combine this with the Quantum Mechanical result:  This tells us that each particle has 2 possible energies: ε+ ≡ - μH for spin “up” ε- ≡ μH for spin “down”  So, for 3 particles, the State of the system is specified by specifying each m =   There are (2)3 = 8 Possible States!!

Possible States of a 3 Spin System in Any One of These 8 States. Given that we know no other information about this system, all we can say about it is that It has Equal Probability of Being Found in Any One of These 8 States.

The system can be only in any one of the states which are However, if (as is often the case in real problems) we have a partial knowledge of the system (say, from experiment), then, we know that The system can be only in any one of the states which are COMPATIBLE with our knowledge. (That is, it can only be in one of it’s accessible states) “States Accessible to the System” ≡ those states which are compatible with all of the knowledge we have about the system. Its important to use all of the information that we have about the system!

(+,+,-) (+,-,+) (-,+,+) Example For our 3 spin system, suppose that we measure the total system energy & we find E ≡ - μH This additional information limits the states which are accessible to the system. Clearly, from the table, Of the 8 states, only 3 are compatible with this knowledge.  The system must be (with equal probability = 1/3) in one of the 3 states: (+,+,-) (+,-,+) (-,+,+)