if E < .l't ( p, q) < E otherwise

Denso QR Bar Code barcode library for javagenerate, create qr code none with java projects

(6.7)

Bar Code generation in javausing barcode integrating for java control to generate, create barcode image in java applications.

It is understood that all members of the ensemble have the same number of particles N and the same volume V. Suppose f(p, q) is a measurable property of the system, such as energy or momentum. When the system is in equilibrium, the observed value of f(p, q) must be the result obtained by averaging f(p, q) over the microcanonical ensemble in some manner. If the postulate of equal a priori probability is to be

Barcode recognizer on javaUsing Barcode decoder for Java Control to read, scan read, scan image in Java applications.

useful, all manners of averaging must yield essentially the same answer. Two kinds of average values are commonly introduced: the most probable value and the ensemble average. The most probable value of f(p, q) is the value of f(p, q) that is possessed by the largest number of systems in the ensemble. The ensemble average of f~p, q) is defined by

Control qr bidimensional barcode size in .net c#to deploy denso qr bar code and qr code 2d barcode data, size, image with visual c# barcode sdk

U)=-------

QR Code 2d Barcode barcode library for .netusing aspx toreceive qr-code for asp.net web,windows application

pd 3Nqf(p, q)p(p, q)

Receive qr code iso/iec18004 on .netgenerate, create qr codes none on .net projects

(6.8)

Control qr barcode data for visual basic.netto develop qrcode and qr data, size, image with visual basic.net barcode sdk

pd 3Nqp(p, q)

Deploy barcode pdf417 for javausing barcode writer for java control to generate, create pdf-417 2d barcode image in java applications.

The ensemble average and the most probable value are nearly equal if the mean square fluctuation is small, i.e., if

Paint bar code on javausing barcode creator for java control to generate, create bar code image in java applications.

(6.9)

STATISTICAL MECHANICS

Control ean / ucc - 14 data for javato use ean 128 and ean 128 barcode data, size, image with java barcode sdk

observed value of f may be calculated. When it is not, we should question the validity of statistical mechanics. In all physical cases we shall find that mean square fluctuations are of the order of liN. Thus in the limit as N ~ 00 the ensemble average and the most probable value became identical. Strictly speaking, systems in nature do not obey classical mechanics. They obey quantum mechanics, which contains classical mechanics as a special limiting case. Logically we should start with quantum statistical mechanics and then arrive at classical statistical mechanics as a special case. We do this later. It is only for pedagogical reasons that we begin with classical statistical mechanics. From a purely logical point of view there is no room for an independent postulate of classical statistical mechanics. It would not be logically satisfactory even if we could show that the postulate introduced here follows from the equations of motion (6.2), for, since the world is quantum mechanical, the foundation of statistical mechanics lies not in classical mechanics but in quantum mechanics. At present we take this postulate to be a working hypothesis whose justification lies in the agreement between results derived from it and experimental facts.

Control qr code iso/iec18004 image for javausing barcode creator for java control to generate, create qr image in java applications.

If this condition is not satisfied, there is no unique way to determine how the

Code 11 encoding on javausing java tobuild usd-8 in asp.net web,windows application

6.2 MICROCANONICAL ENSEMBLE

Word Documents upc a generation with word documentsusing barcode printer for word documents control to generate, create upc-a image in word documents applications.

In the microcanonical ensemble every system has N molecules, a volume V, and an energy between E and E + ~. It is clear that the average total momentum of the system is zero. We show that it is possible to define quantities that correspond to thermodynamic quantities. The fundamental quantity that furnishes the connection between the microcanonical ensemble and thermodynamics is the entropy. It is the main task of this section to define the entropy and to show that it possesses all the properties attributed to it in thermodynamics. Let f ( E) denote the volume in f space occupied by the microcanonical ensemble:

Create qr code jis x 0510 with visual basic.netuse .net windows forms crystal qr code 2d barcode creation togenerate qr code in vb.net

f(E) ==

Qrcode implementation in word documentsuse office word qr code 2d barcode writer toattach qr code jis x 0510 with office word

d 3Npd 3Nq

(6.10)

Make gtin - 128 on .netusing asp.net website toproduce ean/ucc 128 for asp.net web,windows application

E<J('(p, q)<E+!:J.

The dependence of f(E) on N, V, and ~ is understood. Let L(E) denote the volume in f space enclosed by the energy surface of energy E:

(6.11)

Then

f(E) = L(E

+ ~) - L:(E)

(6.12) (6.13)

is so chosen that

E, then

f(E) = w(E)~

CLASSICAL STATISTICAL MECHANICS

where w ( E) is called the density of states of the system at the energy E and is defined by

w(E)

The entropy is defined by

a[(E) aE

(6.14)

8(E, V) == k log f(E)

(6.15)

where k is a universal constant eventually shown to be Boltzmann's constant. To justify this definition we show that (6.15) possesses all the properties of the entropy function in thermodynamics, namely,

( a) 8 is an extensive quantity: If a system is composed of two subsystems

whose entropies are, respectively, 8 1 and 8 2 , the entropy of the total system is 8 1 + 8 2 , when the subsystems are sufficiently large. ( b) 8 satisfies the properties of the entropy as required by the second law of the thermodynamics. To show the extensive property, let the system be divided into two subsystems which have N 1 and N2 particles and the volumes V1 and V2 , respectively. * The energy of molecular interaction between the two subsystems is negligible compared to the total energy of each subsystem, if the intermolecular potential has a finite range, and if the surface-to-volume ratio of each subsystem is negligibly small. The total Hamiltonian of the composite system accordingly may be taken to be the sum of the Hamiltonians of the two subsystems:

.Yt(p, q) = .Yt1(P1' q1)

+ .Yt2(P2' q2)

(6.16)

where (P1' q1) and (P2' q2) denote, respectively, the coordinates and momenta of the particles contained in the two subsystems. Let us first imagine that the two subsystems are isolated from each other and consider the microcanonical ensemble for each taken alone. Let the energy of the first subsystem lie between E 1 and E 1 + ~ and the energy of the second subsystem lie between E 2 and E 2 + ~. The entropies of the subsystems are, respectively, 81(E 1, V1) = k log f 1(E 1)

82(E2, V2) = k log f 2(E2) where f 1(E 1 ) and f 2 (E2 ) are the volumes occupied by the two ensembles in their respective f spaces. They are schematically represented in Fig. 6.1 by the volumes of the shaded regions, which lie between successive energy surfaces that differ in energy by ~. Now consider the microcanonical ensemble of the composite system made up of the two subsystems, and let the total energy lie b'etween E and E + 2~.

"For simplicity we assume that the same N1 , N z particles are always confined respectively to the volumes Vi' Vz . The proof is therefore invalid for a gas, for which S has to be modified (See Section 6.6).