Lecture 22–23 — The Microcanonical Ensemble and the Structure of Thermodynamics#
Source: NPTEL Classical Physics, Mod-01 Lec-22 (The microcanonical ensemble) and Lec-23 (Thermodynamics), Prof. V. Balakrishnan.
Lecture 20–21 split an isolated system into two weakly interacting parts \(A\) and \(B\) and, from the single postulate that all accessible microstates of the whole are equally probable, arrived at
This lecture extremizes that expression. What falls out is not just “the most probable energy split” — it is temperature, entropy, and essentially the entire structure of classical thermodynamics, derived rather than assumed.
The density of states for a free particle#
Before extremizing \(P(E)\), it helps to know how \(\Omega(E)\) actually grows with \(E\). For a single free particle of mass \(m\) confined to volume \(V\), the number of accessible microstates with energy up to \(\varepsilon\) is the phase-space volume of that region, measured in cells of size \(h^3\):
Differentiating gives the density of states, \(\rho(\varepsilon) \equiv d\phi/d\varepsilon \propto \varepsilon^{1/2}\) — the number of microstates per unit energy interval, a quantity that reappears constantly from here through the rest of statistical mechanics. The exponents have a clean origin: the power \(3/2\) splits into a factor of \(3\) from the number of spatial dimensions (momentum space is a \(3\)-sphere; its surface “area” scales as \(p^2\), i.e. \(\varepsilon^1\), before the extra \(1/2\) below) and a factor of \(1/2\) from the quadratic dispersion relation \(\varepsilon = p^2/2m\) — the non-relativistic limit of \(\varepsilon^2 = c^2p^2+m^2c^4\) once the rest energy \(mc^2\) is dropped. In general spatial dimension \(d\), \(\phi(\varepsilon)\propto\varepsilon^{d/2}\) and \(\rho(\varepsilon)\propto\varepsilon^{d/2-1}\):
Three dimensions gives ordinary \(\sqrt\varepsilon\) growth; two dimensions gives a constant density of states, the reason electrons confined to a plane (as in the quantum Hall effect) behave so differently from an ordinary 3D gas; one dimension gives a density of states that diverges as \(\varepsilon\to0\) — low-energy states become more crowded, not less. For \(N\) weakly-interacting particles sharing the total energy roughly evenly, \(\Omega(E)\) compounds this into \(E\) raised to a power proportional to \(N\) — an astronomically increasing function of energy, exactly as Lecture 20–21 argued on general grounds.
Extremizing \(P(E)\): the statistical definition of temperature#
With \(\Omega(E)\) a rapidly increasing function and \(\Omega'(E_{\rm total}-E)\) a rapidly decreasing one, their product is sharply peaked, and locating the most probable macrostate means extremizing \(P(E)\) — equivalently, since these are astronomically large numbers, extremizing \(\ln P(E)\):
using \(E' = E_{\rm total}-E\) so that \(\partial/\partial E = -\partial/\partial E'\). The left side depends only on properties of \(A\), the right side only on properties of \(B\) — two systems that can be physically completely different (a jar of oil in equilibrium with the surrounding air) — yet in equilibrium this one combination matches across the boundary. That combination is defined to be the (inverse) temperature:
Note the direction of dependence: statistical mechanics makes \(T\) a function of \(E\), the reverse of the usual thermodynamic habit of treating \(E\) (or \(U\)) as a function of \(T\). Nothing in this definition requires \(\Omega(E)\) to increase monotonically forever — systems with a bounded energy spectrum can have \(\Omega(E)\) rise and then fall, giving a region where \(\beta<0\), i.e. negative absolute temperature — a real phenomenon in certain spin systems, invisible to the free-particle intuition where \(\Omega(E)\propto E^{3N/2}\) climbs without bound.
Repeating the same argument for exchange of volume and particle number (rather than energy) across the \(A\)–\(B\) boundary gives two more matching conditions, identified as equal pressure and equal chemical potential in equilibrium. All three conditions are partial derivatives of a single function, which is given a name because it — not \(\Omega\) itself — is what appears in every physical formula: the entropy,
From statistical entropy to the first law#
Written in terms of \(S\), the three matching conditions become \(\partial S/\partial E|_{V,N} = 1/T\), \(\partial S/\partial V|_{E,N}=P/T\), \(\partial S/\partial N|_{E,V}=-\mu/T\) — the entropy representation,
equivalently the familiar energy representation \(dE = T\,dS - P\,dV + \mu\,dN\), with \(T=\partial E/\partial S|_{V,N}\), \(P=-\partial E/\partial V|_{S,N}\), \(\mu=\partial E/\partial N|_{S,V}\). This is the first law of thermodynamics — not assumed, but read off directly from the statistical definition of entropy and the equilibrium conditions derived above.
Extensivity, Euler’s relation, and Gibbs–Duhem#
The first law only ever gives increments \(dE\) — it says nothing about the absolute value of \(E\) itself. That extra input is extensivity: in the thermodynamic limit (\(N,V\to \infty\) at fixed density), \(E\) is assumed to be a homogeneous function of degree 1 in the extensive variables \(S,V,N\) — doubling all three at once exactly doubles \(E\). Euler’s theorem for a degree-1 homogeneous function then gives immediately
Two useful facts fall out at once. First, \(E - TS + PV = \mu N\), and the left side is exactly the Gibbs free energy \(G\) (defined via a Legendre transform below) — so \(\mu = G/N\), the chemical potential is simply the Gibbs free energy per particle. Second, differentiating the Euler relation and subtracting the first law term by term leaves
the Gibbs–Duhem relation (\(v=V/N\), \(s=S/N\) the specific volume and entropy) — it says \(\mu\) is a function of the intensive pair \((P,T)\) alone, and generalizes to \(\sum_i X_i\,dF_i = 0\) for any set of conjugate force–flux pairs \((F_i,X_i)\) satisfying \(E=\sum_i F_iX_i\). Both relations depend on extensivity actually holding — self-gravitating systems, where the interaction energy is not short-ranged in the sense Lecture 20–21 assumed, are a standard example where it fails.
The thermodynamic potentials as Legendre transforms#
\(E(S,V,N)\) is inconvenient whenever \(S\) is hard to control directly — exactly the kind of situation Lecture 10–12’s Legendre transform was built for, trading an inconvenient independent variable for its conjugate slope. Applied repeatedly to \(E\):
Potential |
Variables |
Definition |
|---|---|---|
Internal energy \(E\) |
\(S,V,N\) |
— |
Enthalpy \(H\) |
\(S,P,N\) |
\(E+PV\) |
Helmholtz free energy \(F\) |
\(T,V,N\) |
\(E-TS\) |
Gibbs free energy \(G\) |
\(T,P,N\) |
\(E-TS+PV\) |
Grand potential \(\Omega_{\rm gr}\) |
\(T,V,\mu\) |
\(E-TS-\mu N = -PV\) |
\(F\) and \(G\) are the two used constantly in practice; the grand potential is the one that pairs with a grand canonical ensemble (fixed \(\mu\) rather than fixed \(N\)) rather than the canonical ensemble of Lecture 24–25. Differentiating a potential with respect to its natural variables always returns another thermodynamic quantity — \(S=-\partial F/\partial T|_{V,N}=-\partial G/\partial T|_{P,N}\), \(V=\partial G/\partial P|_{T,N}\), \(\mu = \partial F/\partial N|_{T,V} = \partial G/\partial N|_{T,P}\) — the reason these are called potentials at all, by direct analogy with a mechanical potential whose gradient is a force. Throughout, quantities come in conjugate pairs \((T,S)\), \((P,V)\), \((\mu,N)\) whose product always has dimensions of energy; the extensive member of each pair is a state variable, the intensive member a field variable (generalizing to \(\vec E\cdot d\vec P\) for a dielectric or \(\vec B\cdot d\vec M\) for a magnet).
Response functions and stability#
Second derivatives of the potentials are physically response functions. From \(dQ=T\,dS\) at constant \(V,N\), the heat capacity is \(C_V \equiv T\,\partial S/\partial T|_{V,N} = -T\,\partial^2F/\partial T^2|_{V,N}\) — a second derivative of \(F\). Thermodynamic stability requires equilibrium states to sit at a genuine minimum of the appropriate potential (Le Chatelier’s principle), which forces convexity: \(C_V\ge0\), \(C_P\ge0\) (with \(C_P>C_V\)), and the isothermal compressibility \(\kappa_T=-\frac1V\partial V/\partial P|_{T,N}=-\frac1V\partial^2G/\partial P^2|_{T,N}\ge0\) — a fluid is never allowed to expand in response to increased pressure. Since every potential is a function of three independent variables, there are three possible pairs of mixed second derivatives per potential, giving the Maxwell relations — up to \(18\) of them across the six potentials above (only \(4\) independent ones if \(N\) is held fixed throughout).
Worked example: the van der Waals correction and its microscopic origin#
The ideal-gas law \(PV=Nk_BT\) ignores that real molecules interact. The empirical van der Waals equation of state,
corrects for this in two ways whose microscopic origin is worth tracing. The intermolecular potential \(V(r)\) between two neutral molecules has a generic shape: a short-range repulsive core and a longer-range attractive tail. The repulsion is almost entirely a quantum effect — the Pauli exclusion principle forbidding two electrons from occupying the same state — empirically modeled as \(\sim 1/r^{12}\); it is what the excluded volume \(b\) in \(V-Nb\) crudely represents. The attraction is the van der Waals force: even a spherically symmetric, non-polar molecule has instantaneous charge fluctuations that momentarily create a dipole moment \(\vec p_1\); that dipole produces a field \(E\sim p_1/r^3\) at a neighboring molecule, inducing a dipole there proportional to that field (\(p_2=\alpha E\)), and the resulting interaction energy \(U\sim p_2 E \sim E^2 \sim 1/r^6\) is always attractive — the origin of the \(1/r^6\) tail in the standard Lennard-Jones (“6-12”) potential, \(V(r) = V_0\big[(a/r)^{12}-(a/r)^6\big]\). The coefficient \(a\) in van der Waals’ equation has a similarly simple origin: treating every molecule as equally affected by every other (the crudest mean-field approximation), the interaction energy per unit volume scales as (density)\(^2 \propto N^2/V^2\). The van der Waals equation is the special, fully degenerate case of the systematic virial expansion \(PV = Nk_BT\big(1+B_2(T)\,N/V+\cdots\big)\), in which every virial coefficient collapses into the single empirical parameter \(a\).
Where this is headed#
Everything above followed from a single postulate about an isolated system’s own energy shell — but real experiments rarely isolate a system this completely; more often a small object sits inside a much larger environment that can freely exchange energy with it. Lecture 24–25 shrinks \(A\) down to exactly that limit, turning the same postulate into the canonical ensemble and the Boltzmann factor \(e^{-\beta\varepsilon}\) that underlies the rest of statistical mechanics.