Seven days Getting Started statistical mechanics-2nd day Ensemble and distribution function (update)

Terms:canonical Ensemble regular ensemble; partition function distribution functions

The day before the review of statistical mechanics required basic mathematical physics background knowledge, today formally entered the first threshold of statistical mechanics. In the study of physical chemistry/chemical thermodynamics, you will hear the two concepts of "ensemble" and "sub-function" many times, in the absence of a deep study of statistical mechanics, these two words are the existence of God: In principle, what experimental data are not required, only according to the micro-component of a given system of pure Hamilton mechanical properties , you can roll out all the thermodynamic quantities! I am attracted to this idea, and I doubt it, if possible, how is this done?

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2nd day Ensemble and distribution function

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Ensemble

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In his preface to "Statistical mechanics", Mr. Graysessex said:

Modern physicists are aware that the statistical mechanics of leaving "ensemble" are primitive, naïve, inactive, hopeless, even meaningless, contradictory, and unworkable.

The importance of the concept of the ensemble can thus be envisaged.

"Ensemble" McQuarrie definition of Ensemble in the original text of the textbook (Gibbs Ensemble):

An ensemble is a(mental or virtual)Collection of a very large number of systems, say \ (\mathscr{a}\), constructed to is a replica on a thermodynamic (MA Croscopic) level of the particular thermodynamic system of interest.For example, suppose the system has a volumeV, containsNMolecules of a single component, and are known to aE. That's, it is a isolated system withN,V, andEFixed. Then the ensemble would has a volume \ (\mathscr{a} v\), contains \ (\mathscr{a} n\) molecules, and has a total energy \ (\ Mathscr{e} = \mathscr{a}e\).Each of the systems in this ensemble is a quantum mechanical system ofNInteracting atoms or molecules in a container of volumeV. The values ofNandV, along with the between the molecules, is sufficient to determine the energy eigenvaluesEJ of the Schrodinger equation along with their associated degeneracies \ (\omega (E_j) \). These energies is the only energies available to theN-body System. Hence The fixed energyEMust be one of theseEJ S and, consequently, there is a degeneracy \ (\omega (E) \).Note that there is \ (\omega (E) \) Different quantum states consistent with the only things we know on our Macrosco Pic System of interest, namely, the values of N, V, and E. Although all the systems in the ensemble is identical from a thermodynamic point of view, they is not necessarily identi Cal on a molecular level. So far we had said nothing on the distribution of the members of the ensemble with respect to the \ (\omega (E) \) Possib Le quantum states.

Mr Graysessex that the most popular description is Mr Tien's "small hotel analogy", namely:

We envision two experienced experts who want to figure out whether to speed up the flow of customers in a crowded downtown restaurant by paying for a second cup of coffee. To do this, they have to figure out how many valuable seats are wasted by customers who slowly sip their empty cups, which should be used. A connoisseur stares at them from the door and calculates the time they eat and drink, and he observes how the "system" of 10 people changes over time. The second connoisseur arrived at the peak of the business, taking photographs from the balcony to the entire dining room. He was then given the "ensemble" of all the systems of 10 people, thus calculating the percentage of customers eating and drinking at that time and getting the same information. The observation made by the second expert is more convenient and can provide more information.

Mr. Shen further explained:

The long-term average of the system is equal to the ensemble average over time.

According to my personal understanding of this metaphor, I think:

A system (i.e. a thermodynamic system) in which particles are constantly moving, that is, the microscopic state of time changes. And in a few moments of the moment the situation "photographed", you can get a lot of (different) microscopic state . In fact, these microscopic states describe the same system , and the macroscopic nature of the system is determined by the statistics (or overall) of all of these microscopic states . Since the microscopic state can be described as a point in the phase space, we put together these "photos" to ignore the time and constitute a "point" of the phase space. All the points in this "point" graph are fully integrated. Another 1, because time is continuous, microscopic particles movement can also be considered continuous, based on the idea that these points should be able to connect to a line (probably a "ball of yarn"), that is "phase orbit", 2, each point is actually the system may be microscopic state, according to the probability the generation of statistics is very reasonable, that is, the average length of time is equal to the ensemble average.

If the understanding is not right, warmly welcome more correct understanding.

If the above description of the ensemble is understood (in fact, it does not give the definition of the so-called "one sentence") that is "the" ensemble of the public, then the following two basic postulates are very easy to understand:

Every system in the "all-same-sex public" ensemble is the same.

"Experimental Observations" (or "time averages") of "statistical equivalent public" conservative mechanics systems are equivalent to their "ensemble averages".

In addition, there are two articles

"equal odds" for the ensemble under equilibrium , each of the possible microscopic motion states appears to have the same probability.

This article is regarded by Mr T.D. Lee as "the only basic postulate in statistical mechanics".

the "Entropy Computing postulate" is a variety of Boltzmann equations, which relate to the relationship between entropy and the microscopic state of the system.

The general public, equal probability and entropy calculation are the three basic independent public axioms of statistical mechanics. Shen

Note: The Liouville equation is a very important topic, and is considered important to be analogous to the Schrodinger equation on quantum mechanics. However, limitations on time and learning depth are not explored in detail for the moment. The annotation here is updated later.

There is a need to distinguish between the usual types of ensemble, before which two concepts are very important. Two major statistical "elements" in statistical mechanics:

The states of the "state density" system are recorded as \ (\omega\), the d\ (\omega\), and the function of the system Hamiltonian (\varepsilon\). Define the density of the states on this basis

\ (\large D (\varepsilon) =\frac{d\omega (\varepsilon)}{d\varepsilon} \)

Jane's degree g is not considered here.

" statistical weight" is the "number density/phase density/probability density" \ (\rho\) in the phase space, which is related to the Liouville equation and is the solution of the Liouville equation.

The following are the three very common concepts that can be distinguished from the main ensemble:

"micro-regular Ensemble" describes the equilibrium nature of the isolated system.

"Regular Ensemble" describes the equilibrium nature of a closed system.

"Ju Zheng" describes the equilibrium nature of the open system.

For the above three and other common ensembles, detailed analysis will be carried out in subsequent installments.

The concept of the ensemble is here for the time being, and here is the distribution function.

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Distribution function

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The position of the distribution function in statistical mechanics is equivalent to the position of wave function in quantum mechanics. They are also called "generative Functions". Shen

"Distribution Function"

Micro-distribution function of micro-regular ensemble

i.e. \ (\large D (\varepsilon) \)

The distribution function of the regular ensemble

\ (\large Z (\varepsilon) = (\frac{e}{nh^3}) ^n \int e^{-\beta\varepsilon}d (\varepsilon) d\varepsilon\)

The giant distribution function of the giant-Regular ensemble

\ (\large \tilde z=exp[\sum\limits_i (z_i z_i)]\)

where \ (\large z_i=e^{\beta\mu_i} \) is the "ease of life" or "absolute activity" of group I . can actually prove \ (\large \beta=\frac{1}{k_bt}\).

Note: The Lagrange multiply factor method determines the physical meaning of each parameter, and derives the Fermi distribution and Bose distribution involved in the mega-regular ensemble, which is not listed here.

The relationship between the regular distribution function and the thermodynamic quantity (conclusion only):

(1) Average of energy

\ (\large e=-\frac{\partial\mathrm{ln}z}{\partial\beta}\)

(2) Average of the generalized force/pressure

\ (\large y_k = \frac{1}{\beta}\frac{\partial\mathrm{ln}z}{\partial x_k}\)

where \ (\large x_k\) is the generalized coordinate. For pressure P, the generalized coordinate is the system volume V

\ (\large P = \frac{1}{\beta}\frac{\partial\mathrm{ln}z}{\partial v}\)

(3) Entropy

\ (\large s=k_b (\mathrm{ln} z-\beta\frac{\partial\mathrm{ln}z}{\partial \beta})

Other thermodynamic quantities can be obtained by the above three-type and thermodynamic definitions and laws.

(4) Helmholtz free Energy

\ (\large f=-\frac{1}{\beta}\mathrm{ln}z\)

(5) Gibbs free energy

\ (\large g=-k_bt[\mathrm{ln}z-\frac{\partial\mathrm{ln}z}{\partial\mathrm{ln}v}_t]\)

(6) enthalpy

\ (\large h=-k_bt[(\frac{\partial\mathrm{ln}z}{\partial\mathrm{ln}\beta}) _v+ (\frac{\partial\mathrm{ln}z}{\ PARTIAL\MATHRM{LN}V}) _t]\)

(7) Equal capacitance heat capacity

\ (\large c_v=k_bt[t (\frac{\partial^2\mathrm{ln}z}{\partial t^2}) _v+2 (\frac{\partial \mathrm{ln}Z}\partial T{}_V)]\ )

(8) Equal pressure heat capacity

\ (\large c_p=k_bt[t (\frac{\partial^2\mathrm{ln}z}{\partial t^2}) _p+2 (\frac{\partial \mathrm{ln}Z}\partial T{}_P)]+ (\frac{\partial s}{\partial \mathrm{ln} V) _t\)

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3rd Day preview: Phase change and renormalization group

Seven days Getting Started statistical mechanics-2nd day Ensemble and distribution function (update)