Typical models of statistical systems in thermostat

Lectures 5, 6. Typical models of statistical systems in thermostat

 

The plan:

1. Canonical distribution.

2. Big canonical distribution.

3. θ-р distribution.

4. Properties of probabilities of distributions, the sums of states of system and free energy.

 

The statistical description recognizes that any complex system in an equilibrium condition possesses stationary states. Solving stationary equations of  Shredinger, can be found dependence of energy of stationary states Ei and number of stationary states (statistical weight) Г from volume V or other external parameter of system х, numbers of particles in system N. It is usual if to not mean complexity of her decision, a problem of quantum mechanics of the isolated system.

The macroscopical system can be in any of these stationary states, i.e. the state of such system will be mixed. Number of the stationary states accessible to system with energy Е at number of particles N and volume V, we have named statistical weight Г(Е,V,N). The problem is reduced to search of probability of distribution of system on these stationary states.

For the loop isolated system the problem has been solved by Toplmens principle, allowed to receive microcanonical distribution of probability. Results of the statistical description of the isolated system with use of microcanonical distribution have been considered.

Microcanonical distribution of probability has allowed to find and a way of calculation of probability of distribution on states of the system which are being interaction with other bodies.

Let's consider a case when the system is few part of the compound isolated loop system. Three models for the description of such systems have been offered.

The first model is an canonical system. It is small system which interaction with surrounding bodies is reduced only to thermal interaction with thermostat. Such interaction allows maintaining temperature of system to equal temperature of thermostat θ. Other interactions (mechanical, diffusion) are not present; therefore the system exchanges energy with thermostat at constant number of particles and constant volume. Received or given energy leads to transition of system from one stationary state in another. Means, the probability of distribution of canonical system is a probability to find out system in different stationary states with different energy.

If for canonical system, i.e. at a preset value of number of particles N and volume V, Shredingers equation is solved and the spectrum of energy Ei and number of states Γ(Ei), it is enough for record of the sum of states of canonical system is found:

 

 

It is convenient to use and such identical form of record of the sum of states:

 

 

where the designation is entered

 

A(θ,V,N)=-θ lnZ

 

for the  magnitude named by Helmholtz energy

Then the probability of distribution of canonical system on energy can be written down in the next ways:

 

 

On the sense these expressions are probability of that energy of canonical system is equal Ei or probability to canonical system to realize any condition with energy Ei.

For probability of distribution of canonical system on states we shall receive:

 

 

It is probability i-th state of canonical system, or the probability of canonical system to appear in one of states with energy Ei.

Both and in probability of a condition and in sum of states of canonical system energy of a condition of system Ei and temperature θ enter through magnitude

 

zi=e-Ei/θ

 

named by Boltzmanns factor.

Arguments of functions ωi(Ei), ω(Ei), Z  and A are independent variable systems θ,V, N. And the temperature is a parameter which is set thermostat, and the volume and number of particles are magnitudes describing system and their values are certain by system.

Probability of distribution of canonical system name canonical distribution, canonical distribution of Gibbs, canonical distribution of probability, probability of Gibbs canonical distribution.

There are other names and at Helmholtz energy A: Helmholtz energy, free energy of Helmholtz, free energy, free energy at constant volume, function of work, isochoric-isothermal potential, isochoric potential. For a designation use also symbol F.

And the sum of conditions of canonical system Z has other names: the statistical sum, the statistical sum of canonical distribution, the sum of states of canonical distribution, the canonical sum of states.

For cases when interaction of system with other bodies cannot be limited only to thermal interaction, the statistical physics offers two more models of the description of interaction of system and thermostat.

If between system and other bodies except for thermal interaction is available also diffusion interaction (material contact) the opportunity of an exchange with thermostat not only energy, but also particles is given to system. It is greater canonical system. Thermostat in this model is both an energy source, and a source of particles that allows supporting not only a constant temperature of small system in, but supports also to constants chemical potential of this system. Thus in the system varies both number of particles, and energy. To system there are accessible stationary states with different energy and different number of particles.

For the big canonical system at the set volume V we solve Shredingers equation for each possible value of number of particles Ni and for each such case we shall receive a spectrum of energy Ei and number of conditions Г(Ei,N). Energy will depend on number of particles Ei=Ei(Ni):

Using the found magnitudes, we write down the sum of states of the big canonical system:

 

 

where μ- chemical potential of system.

It is useful to enter here again through the sum of states the magnitude named by Landaus energy:

 

Ω(θ,μ,V)=-θ ln Zμ

 

Then the sum of states of the big canonical system will copy so:

 

Zμ=e-Ω/θ

 

Let's write down through energy of Landau and probability of distribution of the big canonical system on energy and particles:

 

It is probability of the big canonical system to contain Ni, particles and to have energy Ei.

For probability of distribution of the big canonical system on states i, we shall receive following formulas:

 

With such probability the big canonical system contains Ni particles and thus is in a state i.

Arguments of functions ωiN(Ei,Ni), ω(Ei,Ni), Zμ and  Ω are independent variables of big canonical system θ, μ, V. The big canonical system defines unequivocally magnitude of volume, and the temperature and chemical potential are set and supported by constants thermostat.

Probability of distribution of the big canonical system name also big canonical distribution of Gibbs, and Gibbs distribution with variable number of particles, and distribution of the big canonical system, θ-μ- distribution.

Energy of Landau Ω  name the even big thermodynamic potential, either potential of Landau, or free energy of Landau, or potential " omega " Ω. It is known, that itself Landau suggested function Ω to name the Supreme potential.

For the sum of states of big canonical system Zμ there are also such names: a big  statistical sum, a big sum, big sum of states, Gibbs sum, the sum of states of Gibbs, the statistical sum of the big canonical distribution, the statistical sum of θ-μ-distribution, the sum of states of θ-μ-system.

Magnitude

 

e(-Ei+μNi)/θ

 

is referse to as Gibbs factor.

Use also magnitude

 

λ=eμ/θ

 

named activity or absolute activity.

Last model is θ-p-system which is applied to the description of thermal and mechanical interaction of small system with surrounding bodies. In this case, thermostat it is allocated with ability to support not only constant temperature, but also constant pressure in system P.

As a result of such interaction with thermostat, the small system with constant number of particles changes the energy Ei, and the volume V (or other parameter x). Possible stationary states of system: dependence of energy Еi on volume Vi and dependence of number of states on energy and volume Г(E,Vi), we find from Shredingers equation with the set number of particles N.

Knowing a spectrum of energy Ei, and number of states Г(E,Vi), we can calculate the sum of states θ-P-systems:

 

 

Let's present ZP in a following kind

 

 

In this record of the sum of states ZP is used the designation for magnitude

 

G(θ,P,N)= -θ ln Zp

 

named Gibbs energy.

Then expression for probability of distribution θ-Р-systems on energy and to volumes will enter the name in a following kind:

 

 

It is probability θ-Р-systems to possess energy Ei, and in volume Vi.

Now the probability of distribution θ-Р-systems on states i and to volumes Vi, will look like:

 

 

In other words, it is probability θ-Р-systems to have volume Vi and to be thus in this i-th state.

Independent variables θ-Р-systems are temperature θ, pressure Р and number of particles in system N. The temperature and pressure for such system are external parameters which are supported, set thermostat. The number of particles in θ- Р system is an internal parameter, it defines considered system.

The received probabilities of distribution θ- Р-systems also have some names: probabilities of θ-Р distribution, θ-Р distribution, the expanded Gibbs distribution, expanded canonical distribution of Gibbs.

The sum of states of θ-Р-systems ZP name also the statistical sum  of θ- Р-distributions and the statistical sum θ-Р-systems, and the sum of states of θ- Р-distribution, and the expanded sum of states, and the sum of states of the expanded canonical distribution.

Historically magnitude G referred to, and sometimes continues to refer to, both Gibbs potential, and thermodynamic Gibbs potential, both simply thermodynamic potential, and free enthalpy, and free energy at constant pressure, both isobaric potential, and isobaric - isothermal potential. For a designation of this function various symbols also were used: Z, F, Ф.

For record of probabilities of stationary states new magnitudes have been entered: Helmholtz energy A(θ,V,N), Landaus energy Ω (θ,μ,V) and Gibbs energy G (θ, P, N). They are unequivocally connected with the corresponding sums of states. All of them have dimension of energy (statistical temperature), are functions of a state of the system and depend on corresponding independent variables. For them there is also a general name - free energy.

The sense of all of this free energy, their communication with the magnitudes used in thermodynamics, becomes clear, when by means of corresponding distributions we shall receive observable magnitudes and thermodynamic ratio. Also the origin of many names of the sums of states and free energy in the same place will clear up. All this is connected by that these concepts have been entered in phenomenological thermodynamics. Free energy together with energy of system name thermodynamic potentials. Later the list of thermodynamic potentials will be added by one more magnitudes - enthalpy. Thermodynamic potentials together with entropy in thermodynamics name characteristic functions.

All the sums of states (the statistical sums) possess property multiplicativity, and all free energy possess property of additivity. We shall remind, that property of additivity possess entropy and energy of system. Additivity is general property of all characteristic functions.

So, the small part of the compound isolated loop system can be described on the basis of simple model, having presented it as the canonical system considering only thermal interaction of system and thermostat. This model enables in addition to consider diffusion interaction (big canonical system) and mechanical interaction (θ-Р-system or the expanded canonical system).

All expressions for probability of distributions on conditions of these systems unite also the general name - Gibbs distributions.

Sometimes the received result formulate as Gibbs theorem: a state of a small part (a small subsystem) microcanonical system are distributed canonically.