3.4.2 Statistical Mechanics#

Prompts

What is the density matrix \(\hat{\rho} = \mathrm{e}^{-\beta \hat{H}}\) and how does it emerge from the imaginary-time propagator?
How do you express thermal averages using the path integral in imaginary time?
Why is the partition function \(Z(\beta) = \text{Tr}(\mathrm{e}^{-\beta \hat{H}})\) equal to a path integral with periodic boundary conditions?
How does treating temperature as inverse imaginary time unify quantum mechanics and statistical mechanics?
If you measure properties of a quantum system at temperature \(T\), does the path integral prediction match experiments?

Lecture Notes#

Overview#

The Wick rotation maps the quantum propagator to a thermal density matrix. This lesson makes that connection precise: the partition function \(Z = \mathrm{Tr}(\mathrm{e}^{-\beta \hat{H}})\) equals a path integral over closed paths in imaginary time (periodic boundary conditions). Thermal averages, free energy, and entropy all follow from a single path integral, and the high-temperature limit recovers classical statistical mechanics in one dimension lower.

Statistical Mechanics via Path Integrals

Statistical mechanics via path integrals reveals a deep unity between quantum mechanics and thermal statistics. The partition function and thermal density matrix emerge as path integrals in imaginary time. The Boltzmann distribution \(\mathrm{e}^{-\beta E}\) is the imaginary-time evolution operator, making temperature and inverse time dual concepts.

The Density Matrix#

In quantum statistical mechanics, the density matrix at temperature \(T\) is

\[\hat{\rho}(T) = \frac{\mathrm{e}^{-\beta \hat{H}}}{Z(\beta)}\]

where:

\(\hat{H}\) is the Hamiltonian
\(\beta = 1/(k_B T)\) is the inverse temperature
\(Z(\beta) = \text{Tr}(\mathrm{e}^{-\beta \hat{H}})\) is the partition function (normalization)

The density matrix is Hermitian (\(\hat{\rho}^\dagger = \hat{\rho}\)) and normalized (\(\text{Tr}(\hat{\rho}) = 1\)). It encodes all statistical information about the system at thermal equilibrium.

Thermal Average#

The thermal average (expectation value at temperature \(T\)) of an observable \(\hat{O}\) is

\[\langle O \rangle_T = \text{Tr}(\hat{\rho} \hat{O}) = \frac{\text{Tr}(\mathrm{e}^{-\beta \hat{H}} \hat{O})}{\text{Tr}(\mathrm{e}^{-\beta \hat{H}})}\]

The Imaginary-Time Propagator as Density Matrix#

The imaginary-time propagator over a time interval \(\beta\hbar\) is

\[K_E(\boldsymbol{x}', \beta\hbar; \boldsymbol{x}) = \langle \boldsymbol{x}' | \mathrm{e}^{-\beta \hat{H}} | \boldsymbol{x} \rangle\]

This is precisely the density matrix in position representation. The path integral over imaginary time from \(\tau = 0\) to \(\tau = \beta\hbar\) gives:

\[K_E(\boldsymbol{x}', \beta\hbar; \boldsymbol{x}) = \int_{\boldsymbol{x}(0)=\boldsymbol{x}}^{\boldsymbol{x}(\beta\hbar)=\boldsymbol{x}'} \mathrm{e}^{-S_E[\boldsymbol{x}]/\hbar} \, \mathcal{D}[\boldsymbol{x}]\]

Diagonal elements (\(\boldsymbol{x}' = \boldsymbol{x}\)):

\[K_E(\boldsymbol{x}, \beta\hbar; \boldsymbol{x}) = \langle \boldsymbol{x} | \mathrm{e}^{-\beta \hat{H}} | \boldsymbol{x} \rangle\]

The local density of states can be extracted from this diagonal element.

Derivation: Partition Function from Imaginary Time

The partition function is the trace of the density matrix:

\[Z(\beta) = \text{Tr}(\mathrm{e}^{-\beta \hat{H}}) = \int d^3\boldsymbol{x} \, K_E(\boldsymbol{x}, \beta\hbar; \boldsymbol{x})\]

Inserting the path integral:

\[Z(\beta) = \int d^3\boldsymbol{x} \int_{\text{periodic in } \tau} \mathrm{e}^{-S_E[\boldsymbol{x}]/\hbar} \, \mathcal{D}[\boldsymbol{x}]\]

The periodicity condition \(\boldsymbol{x}(\beta\hbar) = \boldsymbol{x}(0)\) arises naturally from the trace (i.e., the integration over initial positions).

Thermal Averages from Path Integral#

To compute a thermal average \(\langle O \rangle = \text{Tr}(\hat{\rho} \hat{O})\):

\[\langle O(\boldsymbol{x}) \rangle_T = \frac{1}{Z(\beta)} \int_{\text{periodic}} O(\boldsymbol{x}(\tau_0)) \mathrm{e}^{-S_E[\boldsymbol{x}]/\hbar} \, \mathcal{D}[\boldsymbol{x}]\]

where \(\tau_0\) is an arbitrary time on the circle (thermal average is independent of the insertion point).

Key insight: To measure an observable at temperature \(T\), insert it into the path integral at any imaginary time \(\tau_0 \in [0, \beta\hbar]\) and integrate over all periodic paths.

Free Energy and Thermodynamic Quantities#

Helmholtz Free Energy#

The Helmholtz free energy is related to the partition function by

\[F(T, V, N) = -k_B T \ln Z(\beta)\]

The path integral provides a direct route to \(F\): compute \(Z\) via the path integral, then take the logarithm.

Entropy and Internal Energy#

\[S = -\left(\frac{\partial F}{\partial T}\right)_{V,N} = k_B(\ln Z + \beta \langle \hat{H} \rangle)\]

\[\langle \hat{H} \rangle = -\frac{\partial \ln Z}{\partial \beta}\]

Thermodynamics from the Partition Function

All thermodynamic quantities follow from the single quantity \(Z(\beta)\), which the path integral computes directly. This is the power of the path integral approach: a single functional integral encodes all thermal properties of a quantum system.

Discussion: Quantum-Classical Duality

The partition function \(Z = \text{Tr}(\mathrm{e}^{-\beta \hat{H}})\) can be written as an imaginary-time path integral. This connects quantum mechanics (via Hamiltonian) to statistical mechanics (via temperature). But is this connection fundamental or merely computational?

The high-temperature limit (\(\beta \to 0\), i.e., \(T \to \infty\)) of the quantum partition function should recover classical statistical mechanics. Can you show that \(Z_{\text{quantum}} \to Z_{\text{classical}}\) in this limit?
Conversely, in the low-temperature limit (\(\beta \to \infty\), i.e., \(T \to 0\)), the partition function is dominated by the ground state: \(Z \approx \mathrm{e}^{-\beta E_0}\). How does this emerge from the path integral formulation?
For a classical particle in a potential at temperature \(T\), the partition function is \(Z = \int \mathrm{e}^{-\beta V(x)} dx\) (ignoring kinetic energy in the partition function measure). For the quantum case, kinetic energy affects the path integral measure. Why does temperature couple to position classically but to both position and momentum quantum mechanically?
In field theory, finite-temperature effects correspond to periodic boundary conditions in imaginary time. Can this idea be extended to mechanical systems (particle in a box)?

High-Temperature Limit (\(\beta \to 0\))#

At high temperature, the density matrix becomes sharply peaked near the position of minimum potential energy:

\[\rho(\boldsymbol{x}', \boldsymbol{x}) \approx \sqrt{\frac{m}{2\pi\hbar^2 \beta}} \mathrm{e}^{-V(\boldsymbol{x})/(k_B T)} \delta(\boldsymbol{x}' - \boldsymbol{x})\]

The system is classical: particles bounce around in the potential, governed by Boltzmann statistics \(\mathrm{e}^{-V/(k_B T)}\).

Low-Temperature Limit (\(\beta \to \infty\))#

At zero temperature, the density matrix projects onto the ground state:

\[\rho(\boldsymbol{x}', \boldsymbol{x}) \to |\phi_0(\boldsymbol{x}')\rangle\langle\phi_0(\boldsymbol{x})|\]

where \(|\phi_0\rangle\) is the ground state. The system is purely quantum: in its lowest energy state, exhibiting zero-point energy and quantum correlations.

Computational Advantage: Monte Carlo Simulations#

The Euclidean path integral \(\int \mathrm{e}^{-S_E} \mathcal{D}[\boldsymbol{x}]\) has a positive weight (measure), making it suitable for Monte Carlo importance sampling:

Sample configurations \(\boldsymbol{x}(\tau)\) with probability \(\propto \mathrm{e}^{-S_E[\boldsymbol{x}]/\hbar}\)
Compute observables \(O(\boldsymbol{x})\) on each sample
Average: \(\langle O \rangle \approx \frac{1}{N_\text{samples}} \sum_i O(\boldsymbol{x}_i)\)

In contrast, the real-time path integral \(\int \mathrm{e}^{\mathrm{i}S} \mathcal{D}[\boldsymbol{x}]\) has rapidly oscillating phase (sign problem), making it difficult to simulate directly.

Practical Importance

Lattice QCD (quantum chromodynamics) simulations rely entirely on the imaginary-time formulation. Computing thermal properties, equation of state, and phase transitions at finite temperature requires the Euclidean path integral and Monte Carlo. This is how physicists study the quark-gluon plasma and early-universe physics.

Quantum-Classical Correspondence#

QM at Temperature \(T\) = Classical Stat Mech in \(d+1\) Dimensions

A \(d\)-dimensional quantum system at temperature \(T\) is equivalent to a \((d+1)\)-dimensional classical statistical mechanics problem:

\[ Z_{\mathrm{QM}}^{(d)}(T) = Z_{\mathrm{classical}}^{(d+1)} \]

The \((d+1)\)th dimension is imaginary time \(\tau \in [0, \beta\hbar]\) with periodic boundary conditions (closed paths). Its spatial extent

\[ L_\tau = \beta\hbar = \frac{\hbar}{k_B T} \]

shrinks as temperature rises. This is why:

High \(T\) (short imaginary-time circle): quantum fluctuations are suppressed; the \((d+1)\)-dimensional system collapses to \(d\) dimensions and classical behavior emerges.
Low \(T\) (long imaginary-time circle): the extra dimension opens up; full quantum fluctuations dominate.

Practical consequence: every tool developed for classical statistical mechanics — transfer matrices, Monte Carlo sampling, renormalization group — applies directly to quantum systems via imaginary time. Lattice QCD and quantum Monte Carlo methods for condensed matter both rest on this duality.

Summary#

Thermal density matrix: \(\hat{\rho} = \mathrm{e}^{-\beta \hat{H}}/Z\) — the imaginary-time propagator evaluated at interval \(\beta\hbar\)
Partition function: \(Z(\beta) = \text{Tr}(\mathrm{e}^{-\beta \hat{H}}) = \int_{\text{periodic}}\mathrm{e}^{-S_E/\hbar}\mathcal{D}[x]\) — periodic paths of period \(\beta\hbar\)
All thermodynamics from \(Z\): free energy \(F = -k_BT\ln Z\), entropy, internal energy — all follow from one path integral
Classical limit (\(\beta\to 0\)): kinetic term vanishes, Boltzmann distribution \(\mathrm{e}^{-V/k_BT}\) recovers
Quantum limit (\(\beta\to\infty\)): path integral projects onto ground state \(\mathrm{e}^{-\beta E_0}\)

Homework#

Density matrix normalization. The density matrix is defined as \(\hat{\rho} = \mathrm{e}^{-\beta \hat{H}}/Z(\beta)\) where \(Z(\beta) = \text{Tr}(\mathrm{e}^{-\beta \hat{H}})\). Show directly that \(\text{Tr}(\hat{\rho}) = 1\), starting from the cyclic property of the trace. Why is this normalization essential for interpreting \(\hat{\rho}\) as a probability distribution?
Thermal average of the Hamiltonian. Show that

\[\langle \hat{H} \rangle_T = -\frac{\mathrm{d} \ln Z}{\mathrm{d}\beta}.\]

Then verify this formula for a quantum harmonic oscillator with \(\hat{H} = \hbar\omega(a^\dagger a + 1/2)\).

Density matrix in the energy eigenbasis. Write the density matrix \(\hat{\rho} = \mathrm{e}^{-\beta \hat{H}}/Z\) in the energy eigenbasis \(\{|n\rangle\}\) where \(\hat{H}|n\rangle = E_n |n\rangle\). What is the diagonal element \(\hat{\rho}_{nn}\)? Interpret its physical meaning: which states have the highest probability at low temperature, and which at high temperature?
Imaginary-time propagator and the density matrix. The imaginary-time propagator is \(K_E(\boldsymbol{x}', \beta\hbar; \boldsymbol{x}) = \langle \boldsymbol{x}' | \mathrm{e}^{-\beta \hat{H}} | \boldsymbol{x} \rangle\). Show that the partition function can be written as

\[Z(\beta) = \int d^3\boldsymbol{x}\, K_E(\boldsymbol{x}, \beta\hbar; \boldsymbol{x}).\]

Explain why the trace involves diagonal elements of the propagator.

Free energy and thermodynamics. The Helmholtz free energy is \(F = -k_B T \ln Z\). Starting from this definition, derive the expressions for entropy

\[S = -\left(\frac{\partial F}{\partial T}\right)_{V,N}\]

and internal energy

\[\langle \hat{H} \rangle = F + TS.\]

Comment: why does the path integral representation of \(Z\) provide a powerful tool for computing thermodynamic functions?

Classical limit of the density matrix. In the high-temperature limit (\(\beta \to 0\)), show that the density matrix in position space becomes approximately

\[\rho(\boldsymbol{x}', \boldsymbol{x}) \approx \mathrm{e}^{-[V(\boldsymbol{x}') + V(\boldsymbol{x})]/2k_B T} \, C(\beta)\]

where \(C(\beta)\) is a normalization. Explain why this limit is “classical”—where is the kinetic energy, and why does the potential dominate?

Matsubara zero mode. In the imaginary-time path integral for the partition function, paths are periodic: \(\boldsymbol{x}(\beta\hbar) = \boldsymbol{x}(0)\). One can decompose the path into a constant “zero mode” \(\boldsymbol{x}_0\) and fluctuations. For a free particle, the zero-mode integral gives a factor

\[\int d^3\boldsymbol{x}_0 \, \mathrm{e}^{-S_E^{\text{free}}/\hbar}.\]

Compute this for a free particle in one dimension and verify that it reproduces the partition function \(Z = (2\pi m k_B T/h^2)^{1/2}\) (up to factors). What does this calculation tell you about the importance of spatial translations at finite temperature?