Canonical ensemble

The canonical ensemble is the ensemble that describes the possible states of a system that is in thermal equilibrium with a heat bath (the derivation of this fact can be found in Gibbs ==Properties==

== Free energy, ensemble averages, and exact differentials ==

• The partial derivatives of the function give important canonical ensemble average quantities: • the average pressure is • and the average energy is \langle E \rangle = F + ST. • Exact differential: From the above expressions, it can be seen that the function , for a given , has the exact differential dF = - S \, dT - \langle p\rangle \, dV . • First law of thermodynamics: Substituting the above relationship for into the exact differential of , an equation similar to the first law of thermodynamics is found, except with average signs on some of the quantities: d\langle E \rangle = T \, dS - \langle p\rangle \, dV . • Energy fluctuations: The energy in the system has uncertainty in the canonical ensemble. The variance of the energy is \langle E^2 \rangle - \langle E \rangle^2 = k T^2 \frac{\partial \langle E \rangle}{\partial T}. == Example ensembles ==

"We may imagine a great number of systems of the same nature, but differing in the configurations and velocities which they have at a given instant, and differing in not merely infinitesimally, but it may be so as to embrace every conceivable combination of configuration and velocities..." J. W. Gibbs (1903) Boltzmann distribution (separable system) If a system described by a canonical ensemble can be separated into independent parts (this happens if the different parts do not interact), and each of those parts has a fixed material composition, then each part can be seen as a system unto itself and is described by a canonical ensemble having the same temperature as the whole. Moreover, if the system is made up of multiple similar parts, then each part has exactly the same distribution as the other parts. In this way, the canonical ensemble provides exactly the Boltzmann distribution (also known as Maxwell–Boltzmann statistics) for systems of any number of particles. In comparison, the justification of the Boltzmann distribution from the microcanonical ensemble only applies for systems with a large number of parts (that is, in the thermodynamic limit). The Boltzmann distribution itself is one of the most important tools in applying statistical mechanics to real systems, as it massively simplifies the study of systems that can be separated into independent parts (e.g., particles in a gas, electromagnetic modes in a cavity, molecular bonds in a polymer). Ising model (strongly interacting system) In a system composed of pieces that interact with each other, it is usually not possible to find a way to separate the system into independent subsystems as done in the Boltzmann distribution. In these systems it is necessary to resort to using the full expression of the canonical ensemble in order to describe the thermodynamics of the system when it is thermostatted to a heat bath. The canonical ensemble is generally the most straightforward framework for studies of statistical mechanics and even allows one to obtain exact solutions in some interacting model systems. A classic example of this is the Ising model, which is a widely discussed toy model for the phenomena of ferromagnetism and of self-assembled monolayer formation, and is one of the simplest models that shows a phase transition. Lars Onsager famously calculated exactly the free energy of an infinite-sized square-lattice Ising model at zero magnetic field, in the canonical ensemble. == Precise expressions for the ensemble ==

The precise mathematical expression for a statistical ensemble depends on the kind of mechanics under consideration—quantum or classical—since the notion of a "microstate" is considerably different in these two cases. In quantum mechanics, the canonical ensemble affords a simple description since diagonalization provides a discrete set of microstates with specific energies. The classical mechanical case is more complex as it involves instead an integral over canonical phase space, and the size of microstates in phase space can be chosen somewhat arbitrarily. Quantum mechanical A statistical ensemble in quantum mechanics is represented by a density matrix, denoted by \hat \rho. In basis-free notation, the canonical ensemble is the density matrix : \hat \rho = \exp\left(\tfrac{1}{kT}(F - \hat H)\right), where is the system's total energy operator (Hamiltonian), and is the matrix exponential operator. The free energy is determined by the probability normalization condition that the density matrix has a trace of one, \operatorname{Tr} \hat \rho=1: : e^{-\frac{F}{k T}} = \operatorname{Tr} \exp\left(-\tfrac{1}{kT} \hat H\right). The canonical ensemble can alternatively be written in a simple form using bra–ket notation, if the system's energy eigenstates and energy eigenvalues are known. Given a complete basis of energy eigenstates , indexed by , the canonical ensemble is: : \hat \rho = \sum_i e^{\frac{F - E_i}{k T}} |\psi_i\rangle \langle \psi_i | : e^{-\frac{F}{k T}} = \sum_i e^{\frac{- E_i}{k T}}. where the are the energy eigenvalues determined by . In other words, a set of microstates in quantum mechanics is given by a complete set of stationary states. The density matrix is diagonal in this basis, with the diagonal entries each directly giving a probability. Classical mechanical In classical mechanics, a statistical ensemble is instead represented by a joint probability density function in the system's phase space, , where the and are the canonical coordinates (generalized momenta and generalized coordinates) of the system's internal degrees of freedom. In a system of particles, the number of degrees of freedom depends on the number of particles in a way that depends on the physical situation. For a three-dimensional monoatomic gas (not molecules), . In diatomic gases there will also be rotational and vibrational degrees of freedom. The probability density function for the canonical ensemble is: : \rho = \frac{1}{h^n C} e^{\frac{F - E}{k T}}, where • is the energy of the system, a function of the phase , • is an arbitrary but predetermined constant with the units of , setting the extent of one microstate and providing correct dimensions to . • is an overcounting correction factor, often used for particle systems where identical particles are able to change place with each other. • provides a normalizing factor and is also the characteristic state function, the free energy. Again, the value of is determined by demanding that is a normalized probability density function: : e^{-\frac{F}{k T}} = \int \ldots \int \frac{1}{h^n C} e^{\frac{- E}{k T}} \, dp_1 \ldots dq_n This integral is taken over the entire phase space. In other words, a microstate in classical mechanics is a phase space region, and this region has volume . This means that each microstate spans a range of energy, however this range can be made arbitrarily narrow by choosing to be very small. The phase space integral can be converted into a summation over microstates, once phase space has been finely divided to a sufficient degree. == See also ==