Canonical ensemble
The canonical ensemble ( NVT ensemble or even Gibbs ensemble after JW Gibbs ) is in statistical physics, a system with a fixed number of particles in a constant volume that can exchange energy with a reservoir and is with this in thermal equilibrium. This corresponds to a system with a predetermined temperature, such as a closed system (no exchange of particles ) in a heating bath ( macroscopic system, which is very much greater than the system under consideration ).
Such an ensemble is described by a canonical state. This is the state of equilibrium, the statistical entropy, taking into account the constraint ( the energy expectation value is set ) by the maximum entropy method is a maximum, wherein the condition with the extrinsic parameters such as volume and number of particles must be compatible.
- 4.1 Free energy for equilibrium states
- 4.2 Free energy as a thermodynamic potential
- 4.3 General definition of the free energy
- 5.1 Maximum entropy
- 5.2 Minimum free energy
- 6.1 variation of the energy
- 6.2 equivalence of ensembles in the thermodynamic limit
- 6.3 variation of entropy, pressure and chemical potential
- 7.1 variational problem
- 7.2 Descriptive derivation
Quantum mechanically
The density operator of the canonical state is defined by
Said to belong to the energy eigenstates of energy and are the canonical partition function by
Is given. The parameter can be interpreted as the inverse temperature:
The trace of an operator is defined as follows :, where an arbitrary complete orthonormal system.
In the energy eigenbasis the corresponding density matrix is diagonal
The canonical partition function can be expressed by means of the micro- canonical partition function:
While the sum is counting over all states, the index counts from only the energy level. The number of different states of a particular energy ( that is, the deterioration degree ) is given by.
Classic
Analogously, the classical canonical state ( phase space density)
With the classical canonical partition function
With
Where for identical particles the factor the multiple counting of indistinguishable particles prevents
And for different types of particles with particle numbers and the factor.
The relationship between the classical phase space integral and the trace in the quantum mechanical partition function can be produced by means of coherent states. This also clarifies why the Planck constant occurs in the classical expression.
The canonical partition function can be expressed by means of the micro- canonical partition function:
Thus, the canonical partition function is the Laplace transform of the microcanonical density of states. Since the Laplacian is clear both functions contain identical information.
Expectation values
Below expectation values of various macroscopic quantities are formed. The index k is canonical. The Hamiltonian depends on the volume and number of particles depends on the partition function of temperature, volume and particle respectively.
Energy
The energy expectation value can be calculated via the partition function
Entropy
The statistical entropy can be expressed by the partition function now
Pressure
The pressure of expectation value is equal to:
Chemical potential
For large systems can also calculate the chemical potential ( the particle is a discrete size, only in the thermodynamic limit can be treated quasi- continuous and derivatives with respect to are possible):
Free Energy
Free energy for equilibrium states
Obviously, the logarithm of the sum plays an important role in the calculation of expected values . Therefore, we define the free energy:
Respectively. using the temperature instead of the parameter:
Free energy as a thermodynamic potential
The free energy of the thermodynamic potential of said canonical state. The above expectation values can now be compactly written as a gradient of the potential:
The total differential of the free energy is thus:
General definition of the free energy
Also for non - equilibrium states can define the free energy, as a functional of the density operator on
Or formed
In equilibrium, with or obtained above equilibrium definition of the free energy:
State with extremal properties
Maximum of the entropy
There are two density operators and, which both deliver the same energy expectation value:
The entropy of the state ( which need not necessarily be a state of equilibrium ) can be calculated using the Gibbs inequality as estimate follows:
Is the density distribution of the equilibrium state of the canonical ensemble. Inserting provides:
It follows:
The canonical ensemble has among all ensembles with the same average energy and fixed volume and number of particles, the largest entropy.
Minimum of the free energy
Here is the general definition of the free energy is used.
A condition that does not correspond to the equilibrium state, but the same energy expectation value provides the following applies:
That is, the free energy is minimal at equilibrium.
Fluctuations
Fluctuation of the power
Since the canonical ensemble not have the energy, but only the energy expectation value is set, certain fluctuations are possible. Below the square of the fluctuation in energy is calculated around their mean:
The first derivative of after can be identified with the energy expectation value:
In this case, the heat capacity has been introduced in the last step. She is a susceptibility which indicates how an extensive quantity (energy) with increasing an intensive variable ( temperature ) changes. The response of the energy to an increase in temperature is correlated to the spontaneous fluctuations in the power (refer to fluctuation-dissipation theorem).
The heat capacity is always positive, since the standard deviation can not be negative.
Also can be brought by means of the power fluctuation of the second derivative of the free energy by the temperature in combination:
Equivalence of ensembles in the thermodynamic limit
The heat capacity and thus the mean square fluctuation is an extensive quantity, ie of the order. Similarly, the energy expectation value of the order. The quotient of variation and average is of order:
For thermodynamic systems with particles the ratio is very small (of the order) and thus concentrates the energy distribution very sharply around the mean (see law of large numbers). In the limiting case of large numbers of particles average energy and the energy value with the highest probability to be the same.
The probability density of the energy (not a particular state at a given energy) is given by where the sum is microcanonical state. During the Boltzmann factor decreases monotonically with the energy density of microcanonical increases monotonically with the energy (for example for traditional ideal gas ) so that the product has a maximum. The energy value with the highest probability is given by
It follows:
In the last step while the microcanonical definition of the inverse temperature was in fact identified as the partial derivative of the microcanonical entropy with respect to the internal energy. Thus applies
Therefore corresponds to the most probable value of the energy the energy value of the microcanonical ensemble.
Expanding the logarithm of the probability density of the energy in a power series we obtain:
This is a Gaussian distribution with the width. The relative width of the order and goes to zero, ie the distribution is a delta function. In the limit of large particle numbers microcanonical and canonical ensemble are identical, with the proviso, that the micro- nano African internal energy of the canonical energy expectation value is equal to (eg, and for the classical ideal gas). Both ensembles will include practically the same areas in the phase space ( or states in the Hilbert space ).
The approximate probability density function is then used to calculate the canonical partition function:
From this, determine the free energy:
The last term can be neglected in the thermodynamic limit, as it is, while the others are. Thus, the free energy associated with the canonical ensemble was on sizes of the microcanonical ensemble and recycled.
Variability of entropy, pressure and chemical potential
The range of variation of entropy is due to the fluctuation in energy and thus bring the heat capacity at:
For the quadratic variation of the pressure yields:
And for the chemical potential:
From the positivity of the variance and the isothermal compressibility follows: and
Deriving the steady-state at a prescribed expected value
Variational problem
The equilibrium state at a prescribed expected value (s) can be regarded as a variation problem and can be derived using the method of Lagrange multipliers. Wanted is the density operator, whose statistical entropy is maximized taking into account constraints:
The expression to be maximized with the constraints ( normalization condition ) and ( fixed expectation value of the operator ). If several expectation values established so are multi-component sizes. So to maximize is the following functional of the density operator:
This gives a stationary solution if the first variation of vanishes.
Was used in the last step the relation.
The and each one VONS form; the above sum describes the rutting, the sum over the insertion is one ( taking advantage of completeness )
The term is for all, equal to 0 if it is equal to 0, or if:
This yields the density operator
The definition of a density operator requests that the track is about 1: it follows the Boltzmann - Gibbs state
From the calculus of variations follows only the steady-state behavior; the maximum with respect to the entropy can be shown with the Gibbs inequality.
If the only information about the system, the result of the above- illustrated canonical state ( is another constraint we obtain the grand canonical state).
Descriptive derivation
The heat bath (index 2) and the system of interest (index 1) have low - energy contact. Together they form an overall system that is completely closed to the outside, and thus has to be written mikrokanonisch.
The total system of the Hamiltonian, the Hamiltonian of the subsystems and the operator 's interaction. The latter is used for equilibration of the component systems, although necessary, under the assumption of weak contact but are neglected and over: ie the interaction energy is much smaller than the energy of the individual systems. Thus true and you look at two virtually independent systems. Then the energy is additive and the density matrix multiplication together. The entropy is also due to additive. Furthermore: and.
The total energy remains constant:
The energy of the heat bath is - fold degenerate, the energy of the coupled system is - fold degenerate. Is the degree of degeneracy of the whole system for energy
In the microcanonical ensemble, each possible base state has the same probability. The probability that the system 1 has the energy is equal to the probability that the heating bath, the energy; this is the ratio of total degeneracy of the energy of the system 1, namely, and the total degeneracy degree:
The probability of the system 1 in a given base state ( quantum number ) can be found with energy:
You logarithms and can the entropy of the heat bath and the total entropy identify:
Since the heat bath is much greater than the coupled system, the total energy of the energy almost exclusively of the bath (minus a relatively small average energy of the system 1 ), and can be developed into a Taylor series, the entropy of the first order in order to:
It was used that the derivative of the entropy with respect to the energy is the inverse temperature (see Microcanonical Ensemble ). As a correction to the above development, ie in the order, the following factor occurs:
Here is the heat capacity of the heat bath. The correction terms can be neglected, as is true due to the size of the heat bath. Therefore it is justified itself in the development of entropy to be limited to the first order. This one uses the logarithmic probability. In addition, the independent terms are still together as a constant
Exponenzieren provides the probability that the system 1 is in the base state for energy:
The constant can be on the normalization condition for probabilities determined directly from the properties of the system 1 ( there must be a minimum energy; adopted the energy spectrum EXTEND up, so the sum would diverge ):
Since one is normally only interested in the system 1, system 2, the heating bath is, only on the temperature.