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A cooling schedule for simulated annealing specifies successive effective temperatures and how many proposals to attempt at each. For example, $T_k=T_0{r}^k$ with $0<r<1$, using the geometric sequence, can define stages of equal chosen length. Changing $T$ changes the transition matrix: if the state probability distribution is ${\mu }_n$, then ${\mu }_{n+1}={\mu }_n{P}_{T_n}$, which generally differs from ${\pi }_{T_{n+1}}$. Time spent at one temperature must be compared with its mixing time before equilibrium intuition is justified.

In the five-state barrier example for simulated annealing, fixed $T_{\ast }>0$ gives a finite irreducible Markov chain, so $4$ is eventually visited with probability one. That unbounded-time statement gives no fixed finite deadline. Over any finite number $N$ of trials starting at $1$, the probability of staying at $1$ throughout isEach factor is the rejection probability at $1$. This is an explicit failure event even when every $T_n$ is positive: the best visited objective function value remains $1$, above the global minimum value $0$. More exploratory proposals can help cross a barrier; cooling before those moves occur can preserve a poor local minimum. Successful asymptotic schedules need additional analysis of the allowed moves and barriers.

Historically, Scott Kirkpatrick, Charles Daniel Gelatt Jr. and Mario P. Vecchi connected statistical mechanics with mathematical optimization in 1983. They adapted the Metropolis–Hastings algorithm to cooling searches with symmetric proposals and illustrated computer-design problems.^[ref]

Entropy is an extensive quantity and a state function of an equilibrium system, measured in $\mathrm{J}{\mathrm{K}}^{-1}$. Differences between equilibrium states are calculated along a reversible thermodynamic process:Here heat entering the system is positive and $T$ is its absolute thermodynamic temperature. The reversible path used for calculation need not be the path actually taken. Dividing the actual heat by a single temperature generally fails for an irreversible process.

For example, put two rigid bodies, each with constant heat capacity $C=100\mathrm{J}{\mathrm{K}}^{-1}$, at $600\mathrm{K}$ and $300\mathrm{K}$ in an otherwise isolated system. Assume each body has a uniform internal temperature, no phase transition occurs, and no work is extracted. Their final thermal equilibrium is at $450\mathrm{K}$, because the internal energy lost by one equals that gained by the other. Using entropy from heat capacity,so $\Delta S_{\mathrm{tot}}=C\log (9/8)=11.78\mathrm{J}{\mathrm{K}}^{-1}$. One body's entropy decreases while the total increases. ^[ref]

In statistical mechanics, Boltzmann's entropy formula counts equally likely microstates compatible with a macrostate; Gibbs entropy handles a specified probability distribution over microstates. If the number of equally likely microstates doubles, entropy increases by $k_B\log 2$. The logarithm makes multiplicative counts additive for independent systems. These are precise statements about states and probabilities, not a universal measure of visible messiness. Calling entropy merely "disorder" obscures the thermal calculation above and hides which microstates are being counted. ^[ref]

The usefulness of entropy is that measurable heat capacities determine its changes, and the Second law of thermodynamics turns those changes into constraints on processes. It also connects macroscopic measurements to microscopic descriptions without requiring us to track every particle.

The ergodic hypothesis states that long-time averages of equilibrium observables along typical trajectories equal their ensemble averages on the accessible phase space with the same conserved quantities. It permits statistical mechanics to predict measured macroscopic quantities from equilibrium distributions rather than tracking a trajectory. In its microcanonical form the measure is on the constant-energy surface. It is an assumption about the relevant dynamics, not a consequence of particles being noninteracting; the calculations below use the equilibrium canonical ensemble at temperature $T$.

A density operator describes both a quantum state prepared in one pure state and a statistical preparation whose members occupy different pure states. If normalized vectors $|{\psi }_i⟩$ occur with probabilities $p_i$, defineThe resulting operator is Hermitian, positive semidefinite and has trace one. Positivity follows because $⟨v|\rho |v⟩={\sum }_i{p}_i|⟨{\psi }_i|v⟩{|}^2\ge 0$. Conversely, the spectral theorem gives such a representation of any finite-dimensional operator with these properties, using its eigenvectors and eigenvalues. The matrix representing $\rho$ depends on the chosen basis; the operator and its predictions do not.

One operator gives every measurement probability. For a projective measurement with orthogonal projections ${\Pi }_a$ and outcomes $a$,For example, the first formula follows by averaging $⟨{\psi }_i|{\Pi }_a|{\psi }_i⟩$ over the preparation. The trace expression is basis independent. A general measurement replaces ${\Pi }_a$ by positive measurement effects that sum to the identity, without changing the probability formula. Thus $\rho$ specifies the statistics accessible to measurements on that system.

The distinction between pure and mixed states is especially transparent in the eigenvalues. A pure state has $\rho =|\psi ⟩⟨\psi |$, ${\rho }^2=\rho$, and $\mathrm{Tr}{\rho }^2=1$. A mixed state has at least two positive eigenvalues and $\mathrm{Tr}{\rho }^2<1$. For a qubit,pure states lie on the Bloch sphere; mixed states lie inside the Bloch ball. The maximally mixed qubit $\rho =I/2$ can be prepared by choosing equally between the two $z$ spin eigenstates, or equally between the two $x$ spin eigenstates. These ensembles are different preparation procedures but give the same density operator. No measurement on the resulting qubit alone can identify which ensemble was used.

quantum coherence must be distinguished from merely having two populated levels. In the $z$ basis, the state $|+x⟩$ hasBoth give equal probabilities for the two $z$ outcomes. A measurement along $x$ gives the positive result with probability one for the first state and one half for the second. The off-diagonal entries encode the relative phase information responsible for this difference. Whether a matrix entry is off diagonal is basis dependent; purity is not.

For an isolated system, each component evolves with the same unitary time-evolution operator. Therefore $\rho (t)=U(t)\rho (0)U^{†}(t)$ andUnitary evolution preserves the eigenvalues and purity of $\rho$. A density operator commuting with a time-independent Hamiltonian operator is stationary. A thermal equilibrium example is the canonical ensemble ${\rho }_{\beta }=e^{-\beta H}/Z$, where $Z=\mathrm{Tr}(e^{-\beta H})$ and $\beta =1/(k_{\mathrm{B}}T)$. expectation values then reproduce quantum statistical mechanics without assigning the system a definite energy eigenstate on every run.

A mixed state can also arise without classical randomness in the joint preparation. For a bipartite quantum system, the reduced density operator ${\rho }_A={\mathrm{Tr}}_B{\rho }_{AB}$ gives every measurement probability accessible to subsystem $A$. For the Bell state $(|00⟩+|11⟩)/\sqrt{2}$, the joint state is pure, whereas ${\rho }_A=I/2$. Its local mixedness reflects entanglement with the inaccessible system; the partial trace discards correlations that matter for joint measurements. More generally, interaction with an environment can suppress off-diagonal entries in a selected basis, producing decoherence while the combined system still evolves unitarily.

Finally, state evolution conditional on a measurement outcome differs from isolated evolution. An ideal projective measurement yielding $a$ updates the state by the Lüders rule $\rho \mapsto {\Pi }_a\rho {\Pi }_a/P(a)$. Discarding the outcome gives ${\sum }_a{\Pi }_a\rho {\Pi }_a$, which generally removes coherence between different eigenspaces. The Von Neumann entropy $-\mathrm{Tr}(\rho \ln \rho )$ quantifies mixedness: it vanishes for pure states and reaches $\ln d$ for $I/d$. Multiplying by $k_{\mathrm{B}}$ gives thermodynamic entropy units. These uses make the density operator a common language for preparation uncertainty, measurement, thermal states and entangled subsystems, while retaining their distinct physical origins.

For one coordinate with Hamiltonian $H={\hat{p}}^2/(2m)+V(\hat{x})$, the propagator is $K(x_f,t_f;x_i,t_i)=⟨x_f|e^{-iHT/\hslash }|x_i⟩$, where $T=t_f-t_i>0$. Divide $T$ into $M$ intervals $\delta t$, use the Trotter product formula to separate kinetic and potential evolution, and insert complete position and momentum bases at each slice. The momentum integral is Gaussian:Multiplying the short-time kernels gives the time-sliced configuration-space path integral,with $x_0=x_i$, $x_M=x_f$. Continuum notation abbreviates this limit as ${\int }_{x_i}^{x_f}\mathcal{D}x{e}^{iS[x]/\hslash }$, where $S=\int dt[m{\dot{x}}^2/2-V(x)]$. The measure includes the displayed normalization; it is not an unweighted count of paths. A small damping or analytic-continuation prescription defines the oscillatory integrals. The paths contribute complex amplitudes, so alternative histories interfere before probabilities are calculated. This is the central difference from a classical statistical mixture of trajectories.^[ref]

When actions vary rapidly relative to $\hslash$, stationary phase selects classical paths. Put $x=x_{\mathrm{cl}}+\eta$, with $\eta$ zero at the endpoints. Vanishing first variation gives $m{\ddot{x}}_{\mathrm{cl}}+V^{\prime }(x_{\mathrm{cl}})=0$. The quadratic term isThe resulting Gaussian fluctuation determinant is the semiclassical propagator prefactor. In one dimension, away from caustics,The branch convention makes $\nu =0$ for a direct free-particle path before any caustic. The Maslov index tracks the phase changes as fluctuations pass through a focal singularity. In several dimensions replace the endpoint derivative by the Van Vleck determinant. The approximation sums all relevant classical paths; real-time action need only be stationary, not a minimum. It fails in this simple form at a caustic or coalescing saddles, where a uniform approximation is needed. Quadratic actions have no higher fluctuation terms, making their Gaussian evaluation exact. For example $S_{\mathrm{cl}}=m(x_f-x_i{)}^2/(2T)$ gives the exact free-particle kernel $\sqrt{m/(2\pi i\hslash T)}{e}^{i{S}_{\mathrm{cl}}/\hslash }$. Tunnelling involves classically forbidden regions and can be described by imaginary-time saddles rather than real classical trajectories.

Quantum statistical mechanics follows by Wick rotation $t=-i\tau$. The imaginary-time evolution operator is $e^{-\tau H/\hslash }$, with Euclidean actionTaking a thermal trace identifies and integrates the endpoint positions:This thermal trace as a closed Euclidean path integral explains the periodic imaginary-time direction. A density-matrix element instead retains open endpoints. For many identical particles the trace also sums over endpoint permutations, with a plus sign for bosons and permutation parity for fermions. In coherent-state language this becomes periodic bosonic fields and antiperiodic Grassmann fields, with corresponding Matsubara frequencies.

The Euclidean weight resembles a classical Boltzmann weight. A time-sliced local quantum field in $d$ spatial dimensions becomes a classical-looking field model in $d+1$ dimensions, with anisotropic couplings and extra extent $\beta \hslash$. Thus zero-temperature quantum criticality can be treated using an infinitely extended imaginary-time direction, while nonzero temperature imposes a finite thickness. The dynamical scaling of time need not equal the spatial scaling: the dynamical critical exponent specifies their relationship. This quantum-to-classical imaginary-time mapping requires qualification when weights have signs or imaginary terms; such a functional integral is not automatically an ordinary positive classical probability distribution. Even with a positive weight, the mapping concerns equilibrium statistics, not literal dissipative real-time trajectories. For a free particle the Euclidean kernel is a diffusion kernel with diffusion coefficient $\hslash /(2m)$, illustrating the mathematical correspondence.

The classical high-temperature limit is a further approximation, rather than simply another name for imaginary time. If quantum level spacings and exchange effects are negligible, a one-particle partition function in $d$ dimensions becomesFor indistinguishable particles there is a factor $1/N!$ in the dilute classical gas, and exchange is negligible when the number density times the thermal de Broglie wavelength to power $d$ is small. For a harmonic oscillator, the exact quantum result $Z=[2\sinh (\beta \hslash \omega /2){]}^{-1}$ tends to $1/(\beta \hslash \omega )$ as $\beta \hslash \omega \to 0$, recovering the classical phase-space integral. stationary phase explains classical mechanics; the short imaginary-time, weak-exchange limit explains classical statistical mechanics. These are related but distinct limits of the same quantum formulation.

Statistical mechanics connects microstates to macrostates through probability distributions. Its microcanonical ensemble, canonical ensemble and grand canonical ensemble specify different constraints on exchanges with the surroundings. An ensemble average describes repeated preparations under those constraints; it need not be the value in every individual preparation.

This makes the subject useful whenever tracking every microscopic detail would obscure the question being asked. To predict a heat capacity, we need how the expected value of energy changes with temperature, rather than a record of every unit's state. The partition function packages that calculation into one reusable object.

In his 1902 book, J. Willard Gibbs systematically developed the canonical ensemble. His preface contrasts following one system through time with studying a collection of systems with different microscopic conditions. Chapter IV defines the canonical ensemble through an exponential function of energy and establishes its normalization. This supplied a general strategy for predicting collective measurements by averaging over possible microscopic conditions, with the partition function organizing the calculation. ^{[ref] [ref]}

Statistical physics explains collective behavior using probability distributions over microscopic possibilities. Its central surprise is that incomplete microscopic knowledge can still produce precise predictions for internal energy, entropy and heat capacity. The bridge is statistical mechanics: specify the available microstates, assign their probabilities, then compute the expected values of measured quantities.

A two-level system already displays the idea. Warming it makes its higher energy more likely; its heat capacity first rises, then falls as the two probabilities approach equality. With two distinguishable units, merely counting microstates explains why the middle energy has twice the state degeneracy. These small examples expose the machinery before the much larger sums needed for a photon gas or an Ising model.

The appeal is economical: the same microscopic energies and probabilities explain both an average reading and its variance. A measurement's spread therefore becomes useful physical information rather than an inconvenience to discard.

A mathematician and physicist associated with statistical mechanics and the Ehrenfest model with Paul Ehrenfest.

Thermal electrical noise is the fluctuating electric potential difference or electric current produced by thermal fluctuations in a dissipative electrical circuit, even without an applied voltage. A resistor at thermal equilibrium is both a source of noise and a sink for electrical energy. Its electrical resistance describes the dissipation, and its absolute temperature sets the scale of the fluctuations. This connects a practical limit on signal-to-noise ratio to statistical mechanics: a warm, unbiased resistor is not electrically silent.

John Bertrand Johnson measured the effect using a vacuum-tube amplifier and thermocouple; his 1928 paper reported agreement with the Boltzmann constant. Harry Nyquist's companion paper calculated the effect from thermodynamics and statistical mechanics.^[ref][ref]