Quantum statistics
Many (if not most of) the collective phenomenon can’t be explained with classical statistics. In fact, beyond simple dilute gas, the explanatory power of classical statistics fail. Actually, we often tend to consider quantum mechanics as an abstract topic, far from our daily life, which concern only very specific phenomenon, as the ones for which there is so few interactions that decoherence can be avoided. In a sense, statistical mechanics disprove this assertion by demonstrating that, without the intrinsic quantum nature governing microphysics, it would be impossible to interpret most of the daily life phenomenon around us from a statistical perspective.
In particular, we absolutely need quantum statistics in order to describe:
- Systems with high densities, like solids or cristals.
- To describe radiation, like light, as the well known UV catastrophe illustrates.
We will assume here some prior knowledge of quantum mechanics and we warmly advise the reader to first have a look at the quantum mechanics class before reading further.
There is no standard and consensual way to go from the classical/continuous description based in phase space to the quantum one in Hilbert space. The transition between both is often explored using semi-classical limits and more or less general quantization receipes. Finding a rigourous and general way to go from phase to Hilbert spaces is an active research area, with promising directions such as the one proposed by geometrical quantization. We will not attempt to discuss this matter in detail here and instead we will present Hilbert space as a distinct object from phase space, used to describe the statistics of quantum systems.
Microstates: Wave-vectors, density operator and entropy
A quantum system is described by a wave-vector $\ket{\phi}$, element of a complex Hilbert space $\mathcal{H}$ equipped with an Hermitian product $\braket{.}{.}: \mathcal{H}\times\mathcal{H}\to \mathbb{C}$. As such, the wave-vector describing the system is the microstate of the system in a quantum mechanical context. Observables are described by Hermitian operators $\hat{f}$ on this Hilbert space, that is object that act on wave-vector to give another wave-vector $\hat{f}: \mathcal{H}\to\mathcal{H}$. The wavevector describing the system can be written as a linear decomposition on the basis of the eigenvectors $\ket{f_i}$ of an operator $\hat{f}$ as:
\[\ket{\phi} = \sum_i \alpha_i \ket{f_i}\]Where the coefficients $\alpha_i= \braket{\phi}{f_i}$ are complex coefficients, the probability amplitudes and, by definition of the eigenvectors:
\[\hat{f}\ket{f_i} = f_i \ket{f_i}\]The probability for the system to be observed in the state $\ket{f_i}$ with the measured value $f_i$ for the observable $\hat{f}$ is then given by $p(f_i) = \alpha_i^* \alpha_i$ (and so $\sum_i \alpha_i^* \alpha_i = 1$) and the average value of $\hat{f}$ is given by $\langle f \rangle = \bra{\phi}\hat{f}\ket{\phi}$. In this sense, $\ket{\phi}$ contains all the quantum informations about the system.
The density operator $\hat{\rho}$, generalizes the wavevector to take into account for more general states, called mixed states $\ket{\psi}$ that are statistical mixtures of pure states $\ket{\phi_i}$.
\[\hat{\rho}:= \ket{\psi(t)}\bra{\psi(t)} = \sum_i \mathcal{P}_i \ket{\phi_i(t)}\bra{\phi_i(t)}\]Where $\mathcal{P}_i$ are the classical probabilities – which can also be time dependent – associated to the pure state $\ket{\phi_i}$.
This operator is necessary in statistical physics where the system is composed of a big number of independant quantum sub-systems, all described by pure states. Despite the quantum nature of all the subsystems, the mixture behaves stochasticly with classical probabilities. It can also be used when the system considered is the sub-part of a quantum entangled system (for example considering a single electron in an intriqued “EPR” electron-positron pair). In such a situation, we can show that this sub-part behaves as a classical statistical variable. As such, the best and more general tool to describe a microstate in the quantum context is given by $\hat{\rho}$.
The mean value of an observable $\hat{f}$ is obtained from $\rho$ using the trace:
\[\langle \hat{f}\rangle = {\rm Tr}(\hat{\rho}\hat{f})\]Complement: More informations on the density operator and the trace
$$Tr(\hat{A}) = \sum_i \bra{\phi_i} \hat{A}\ket{\phi_i}$$Out of the density operator, one can build the Von Neumann entropy:
\[\boxed{S = -\text{Tr}(\hat{\rho} \ln(\hat{\rho}))}\]which is equivalent to the classical entropy. Due to its concavity, extremalization is maximization. Using the Lagrange multiplier technique we find the expression for $\hat{\rho}$ maximizing $S$.
The constraints for maximizing $S$ are normalization of all probabilities and the mean value of an observable $\hat{X}$, which takes the form:
\[\begin{aligned} \begin{cases} \text{Tr}{(\hat{\rho})}&=1\\ \text{Tr}{(\hat{\rho}\hat{X_i})}&= \langle X_i \rangle \end{cases} \end{aligned}\]After the discrete and the continuous case, we get a third flavour of the exact same expressions for the ensembles by maximising the entropy, with very similar prooves based on the Lagrange multipliers technique.
Microcanonical ensemble: maximising $S$ using only the normalisation of probabilities we find
\[\rho = \frac{1}{N}\mathbb{I}\]where $N$ is the dimension of the Hilbert space and $\mathbb{I}$ is the unit operator on $\mathcal{H}$ such that $\ket{\psi}=\mathbb{I}\ket{\psi}$ for all $\psi \in \mathcal{H}$.
Proof
In this case, we have no information about the system other than it must exist in a valid state. The only constraint is the normalization of the density operator: $\text{Tr}(\hat{\rho})=1$ Using Lagrange multipliers: $$\mathcal{L}(\hat{\rho})=−\text{Tr}(\hat{\rho}\ln(\hat{\rho})-\alpha(\text{Tr}(\hat{\rho})−1)$$ Taking the functional derivative with respect to $\hat{\rho}$ and setting it to zero: $$\frac{\delta \hat{\rho}\delta}{\delta \hat{\rho}}=−\ln(\hat{\rho})−\mathbb{I}−\alpha\mathbb{I}=0$$ $$ln(\hat{\rho})=−(1+\alpha)\mathbb{I}$$ $$\hat{\rho}=e^{−(1+\alpha)}\mathbb{I}$$ Since $e^{−(1+\alpha)}$ is a constant, we call it $1/N$. Normalization $Tr(\hat{\rho})=1$ implies $N$ is the dimension of the accessible Hilbert space, yielding the equiprobability state: $$\rho = \frac{1}{N}\mathbb{I}$$(Grand) canonical ensemble: considering now a system that can exchange both particles and energy with the surrounding medium at equilibrium. The number of particle $N$ is an observable, and we thus assume that there exist an operator $\hat{\mathcal{N}}$ on $\mathcal{H}$ associated to the number of particles in the system. We will discuss in the next lecture exactly how to provide such an operator. Adding conditions on the mean values of energy and particle number in the entropy maximisation, we get the quantum generalisation of the Grand canonical ensemble:
\[\boxed{\hat{\rho}= \frac{1}{\Xi}e^{-\beta(\hat{H} - \mu \hat{\mathcal{N}})}}\]With $\Xi$ is the quantum partition function:
\[\Xi = \text{Tr}(e^{-\beta(\hat{H} - \mu \hat{\mathcal{N}})})\]Without surprise, we find here, with an identical proof, the same expressions for the ensemble distributions of discrete and classical statistical mechanics (the canonical ensemble can be found using only constraint on energy, leading to $\mu=0$) and all the ensembles have identical structure to the classical ones. However, we put more emphasize on the grand canonical ensemble here than in classical physics, as – in a quantum context– particles can be created and destroyed even within a closed box! Indeed think for exemple about photons being absorbed or emitted by atoms. As such, the grand canonical model is the minimal ensemble we want to consider in a quantum context.
Proof
In Fock space, we impose a constraint on the mean particle number $\langle \mathcal{N} \rangle = \mathrm{Tr}(\hat{\rho} \hat{\mathcal{N}})$ and energy $U = \mathrm{Tr}(\hat{\rho} \hat{H})$. We then define the Lagrangian to maximize the entropy under constraints: $$ \begin{equation} \mathcal{L} = - \mathrm{Tr}(\hat{\rho} \ln \hat{\rho}) - \lambda_0 \left( \mathrm{Tr}(\hat{\rho}) - 1 \right) - \beta \left( \mathrm{Tr}(\hat{\rho} \hat{H}) - \langle E \rangle \right) - \gamma \left( \mathrm{Tr}(\hat{\rho} \hat{\mathcal{N}}) - \langle N \rangle \right), \end{equation} $$ where $\lambda_0, \beta, \gamma$ are Lagrange multipliers. Taking the functional derivative of $\mathcal{L}$ with respect to $\hat{\rho}$ and setting it to zero gives: $$ \begin{equation} \hat{\rho} = e^{-(\lambda_0 + 1)} \, e^{-\beta \hat{H} - \gamma \hat{\mathcal{N}}}. \end{equation} $$ We identify the chemical potential $\mu$ by $$ \begin{equation} \gamma = - \beta \mu, \end{equation} $$ and define $\Xi$ as $$ \begin{equation} \Xi = e^{\lambda_0 + 1} = \mathrm{Tr} \left( e^{-\beta (\hat{H} - \mu \hat{\mathcal{N}})} \right). \end{equation} $$Quantum phase-space: time evolution and Ehrenfest theorem
The classical phase space of classical mechanics is replaced by the Hilbert space $\mathcal{H}$ in a quantum context. As discussed above, observables $\hat{f}$ are not functions like in classical mechanics but operators $\hat{f}:\mathcal{H}\to\mathcal{H}$. Just like in classical mechanics, a special observable, the Hamiltonian $\hat{H}$ governs the time evolution of microstates, through the Schrödinger equation:
\[i\hbar \frac{\text{d}}{\text{d}t}\ket{\phi} = \hat{H}\ket{\phi}\]which solution gives the time evolution of the wave-vector from a time $t_0$ to a time $t$ as
\[\ket{\phi(t)}= U(t-t_0)\ket{\phi(t_0)}\]with the time evolution operator:
\[U(t-t_0)=e^{\frac{-i \hat{H}t}{\hbar}}\]As in classical mechanics, the commutator with $\hat{H}$ allows to infer the time evolution of the mean value of observables through the Erhenfest theorem, which follows from the Schrödinger equation:
\[\frac{\text{d}}{\text{d}t} \langle \hat{f} \rangle= \frac{1}{i\hbar}\langle [\hat{H},\hat{f}] \rangle +\langle \frac{\partial \hat{f}}{\partial t}\rangle\]Proof:
$$ \begin{align} \frac{\text{d}}{dt} \langle \hat{f} \rangle &= \frac{\text{d}}{\text{d}t} \bra{\phi(t)} \hat{f}(t) \ket{\phi(t)} \\ &= \left(\frac{\text{d}}{\text{d} t}\bra{\phi(t)}\right) \hat{f}(t) \ket{\phi(t)} + \bra{\phi(t)} \frac{\partial \hat{f}(t)}{\partial t} \ket{\phi(t)} + \bra{\phi(t)} \hat{f}(t) \frac{\text{d}}{\text{d} t} \ket{\phi(t)}. \end{align} $$ Using Schrödinger equation: $$ \begin{align} \begin{cases} i \hbar \frac{d}{dt} \ket{\phi(t)} &= \hat{H} \ket{\phi(t)}, \\ -i \hbar \frac{d}{dt} \bra{\phi(t)} &= \bra{\phi(t)} \hat{H}, \end{cases} \end{align} $$ we get: $$ \begin{align} \frac{d}{dt} \langle \hat{f} \rangle &= \frac{1}{i \hbar} \bra{\phi(t)} \hat{H} \hat{f}(t) \ket{\phi(t)} - \frac{1}{i \hbar} \bra{\phi(t)} \hat{f}(t) \hat{H} \ket{\phi(t)} + \bra{\phi(t)} \frac{\partial \hat{f}(t)}{\partial t} \ket{\phi(t)} \\ &= \frac{1}{i \hbar} \langle [\hat{f}(t), \hat{H}] \rangle + \left\langle \frac{\partial \hat{f}(t)}{\partial t} \right\rangle. \end{align} $$The geometry of the Hilbert space is encoded in the operator commutator $[.,.]$, which is the quantum equivalent to the classical Poisson bracket:
\[[\hat{f},\hat{g}]= \hat{f}\hat{g} -\hat{g}\hat{f}\]It disctates the time evolution of $\hat{\rho}$ (which is time depend through $\mathcal{P}_i(t)$ and $\ket{\phi_i(t)}$):
\[\frac{\text{d}\hat{\rho}}{\text{d}t}= \frac{1}{i\hbar} [\hat{H},\hat{\rho}]\]which follows and generalizes the Schrödinger equation.
Proof:
The probability distribution is independant of time if $\hat{H}$ commutes with $\hat{\rho}$: $[\hat{H},\hat{\rho}]=0$. This condition is satisfied for any function of the Hamlitonian: $\hat{\rho}=f(\hat{H})$, which are characteristic of thermal equilibrium.
Multiple particles
All this is nice and well. We motivated the use of Hilbert space instead of phase space as a necessary transition to go from classical systems to quantum systems. The two spaces share deep similarities in the form of their equations (time evolution of observable, entropy, partition function), as well as profound disimilarities in their internal structures and geometry (functions/operators, Poisson bracket/commutators). Now, in order to do proper statistical physics, we must understand what exactly should be the wave-vector $\ket{\phi}$, or the density operator $\hat{\rho}$ in the case of a system composed of multiple particles. This will be the topic of our next lecture.