From Dynamics to Probability: Microstates and Ensembles
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You should be comfortable with Newtonian mechanics for many-particle systems and understand the basic definitions of temperature, pressure, and internal energy from thermodynamics.
Why this matters
Thermodynamics tells us what happens, but statistical mechanics explains why it happens.
The transition from deterministic dynamics to probabilistic descriptions is the foundational shift in thermal physics. While classical mechanics provides exact trajectories for every particle, the sheer number of degrees of freedom in macroscopic systems makes tracking them impossible. Instead, we describe the system by the probability of finding it in a particular configuration. This lecture establishes the conceptual framework: defining what a microstate is, why we must group them into macrostates, and how the choice of ensemble determines which thermodynamic variables are fixed. This sets the stage for the microcanonical ensemble, where we will first encounter the statistical definition of entropy.
The Bridge Between Micro and Macro
Imagine a gas in a box. Thermodynamics describes it using a few variables: pressure, volume, and temperature. However, the box contains roughly \(10^{23}\) molecules, each with a position and a velocity. The exact state of all these molecules is called a microstate. Because we cannot measure or control every single molecule, we do not care about the specific microstate. Instead, we care about the macrostate, which is defined by the coarse-grained variables we can measure, such as the total energy or the volume.
The key insight is that many different microstates correspond to the same macrostate. For example, a gas at a specific temperature and volume can have its molecules arranged in countless ways, yet it looks the same to a thermometer. Statistical mechanics assumes that all accessible microstates are equally likely. The probability of a macrostate is then proportional to the number of microstates that fit its description. This shift from tracking individual particles to counting possibilities is the core of the subject.
Phase Space and the Liouville Theorem
To formalize this, we use the concept of phase space. For a system of \(N\) particles, the phase space is a \(6N\)-dimensional space where each point represents a unique microstate. The coordinates are the positions \(\mathbf{r}_i\) and momenta \(\mathbf{p}_i\) for each particle \(i\).
A fundamental result from classical mechanics, the Liouville theorem, states that the density of points in phase space is conserved along the trajectories of the system. In simpler terms, if you have a "cloud" of possible states, its volume in phase space does not change as the system evolves. This ensures that the probability distribution of the system remains consistent over time, allowing us to define a steady-state probability density \(\rho(\mathbf{r}, \mathbf{p})\). Liouville's theorem allows a stationary distribution, but it does not by itself prove that time averages equal ensemble averages; that further assumption is the ergodic hypothesis.
Here, \(\rho\) is the phase space density. The equation indicates that the probability density is constant along the flow of the system in phase space.
Defining the Ensembles
An ensemble is a collection of a large number of identical systems, each in a different microstate, but all consistent with the same macroscopic constraints. We do not physically build these ensembles; they are a mathematical tool to calculate probabilities.
There are three primary ensembles used in statistical mechanics, distinguished by which thermodynamic variables are held fixed:
- Microcanonical Ensemble: The system is isolated. The total energy \(E\), volume \(V\), and number of particles \(N\) are fixed. This is the most fundamental ensemble.
- Canonical Ensemble: The system is in contact with a heat bath. The temperature \(T\), volume \(V\), and number of particles \(N\) are fixed. The energy fluctuates.
- Grand Canonical Ensemble: The system is in contact with a heat and particle bath. The temperature \(T\), volume \(V\), and chemical potential \(\mu\) are fixed. Both energy and particle number fluctuate.
The choice of ensemble depends on the physical setup. For an isolated system, we use the microcanonical ensemble. For a system in a thermostat, we use the canonical ensemble. The results from all three ensembles are equivalent in the thermodynamic limit (where \(N \to \infty\)), but they offer different mathematical tools for different problems.
The Role of the Partition Function
In the canonical ensemble, the central quantity is the partition function, denoted by \(Z\). It is a sum over all possible microstates of the system, weighted by the Boltzmann factor \(e^{-\beta E_i}\), where \(\beta = 1/(k_B T)\) and \(E_i\) is the energy of the \(i\)-th microstate.
The partition function acts as a normalization constant for the probability distribution. The probability of finding the system in a microstate with energy \(E_i\) is given by \(P_i = e^{-\beta E_i} / Z\). All thermodynamic quantities, such as the internal energy \(U\), entropy \(S\), and free energy \(F\), can be derived from \(Z\) and its derivatives. For example, the internal energy is the average energy: \(U = -\frac{\partial \ln Z}{\partial \beta}\). This demonstrates the power of the partition function: it encapsulates all the thermodynamic information of the system.
A common misconception is that the microcanonical ensemble is the only "true" ensemble because it deals with isolated systems. In reality, the canonical ensemble is often more practical because most real systems are not perfectly isolated. The equivalence of ensembles holds only in the thermodynamic limit. For small systems, such as nanoscale particles, the fluctuations in energy and particle number are significant, and the choice of ensemble can lead to different predictions. C
Worked example
Problem. A box holds 100 gas molecules, each equally likely to be found in the left or right half. A microstate lists which half each molecule is in; a macrostate gives only the number \(n_L\) in the left half. Find the number of microstates for \(n_L = 50\) and for \(n_L = 100\), the probability of each macrostate, and the probability that \(n_L\) lies between 45 and 55.
- Total number of microstates: \(2^{100} = 1.27 \times 10^{30}\), all equally likely.
- Microstates with \(n_L = 50\): \(\Omega(50) = \binom{100}{50} = 1.01 \times 10^{29}\).
- Microstates with all molecules on the left: \(\Omega(100) = 1\).
- Probabilities: \(P(50) = 1.01 \times 10^{29}/1.27 \times 10^{30} = 0.080\), while \(P(100) = 2^{-100} = 7.9 \times 10^{-31}\).
- Summing \(\binom{100}{n}/2^{100}\) for \(n = 45\) to 55 gives 0.729: almost three-quarters of the time the split is within 5 of even, even for only 100 molecules. For \(10^{23}\) molecules the relative spread shrinks as \(1/\sqrt{N}\), to a few parts in \(10^{12}\).
Units check. Numbers of microstates and probabilities are pure numbers; probabilities lie between 0 and 1.
Three quick tests
- What is the main difference between the microcanonical and canonical ensembles?
Answer
The microcanonical ensemble fixes energy, while the canonical ensemble fixes temperature. - Why is the Liouville theorem important for statistical mechanics?
Answer
It ensures that the phase space volume is conserved, justifying the use of probability distributions. - What is the physical meaning of the partition function \(Z\)?
Answer
It is a sum over all microstates weighted by their Boltzmann factors, acting as a normalization constant for probabilities.
Problem set for PU-203 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes checked line by line by the editor; every number in the worked example was recomputed.