统计力学精要：加拉沃蒂的短篇论述

5星 · 超过95%的资源需积分: 9 75 浏览量更新于2024-08-02 收藏 2.45MB PDF 举报

"本书《统计力学》由Giovanni Galavotti撰写，是关于统计力学的简明论著，源自作者在意大利物理学百科全书统计力学部分的工作。内容包括了他在百科全书中撰写的多篇论文，特别是关于‘统计力学’、‘集合论’和‘布朗运动’的文章。该书在意大利版的基础上进行了修订和扩充，由罗马‘La Sapienza’大学物理系出版，并得到了意大利国家研究委员会(CNR-GNFM)的支持。" 统计力学是物理学的一个分支，它通过统计方法来研究大量粒子系统的宏观性质和行为。这个领域起源于19世纪末，由詹姆斯·克拉克·麦克斯韦、路德维希·玻尔兹曼和阿尔弗雷德·埃伦费斯特等人发展起来，目的是将热力学定律与微观世界的牛顿力学联系起来。本书的核心概念包括： 1. **微观状态与宏观状态**：统计力学的基础是理解一个系统的宏观性质（如温度、压力和体积）是由其所有粒子的微观状态决定的。这些微观状态的数量巨大，统计力学通过统计方法来处理。 2. **熵和概率**：熵是描述系统混乱程度的物理量，与热力学第二定律密切相关。在统计力学中，熵与微观状态的概率分布有关，最大熵原理指导我们如何确定最可能的宏观状态。 3. **玻尔兹曼统计**和**费米-狄拉克统计**/**玻色-爱因斯坦统计**：这三种统计方法分别适用于经典系统、费米子系统和玻色子系统。它们决定了不同能级上的粒子分布，影响系统的热力学性质。 4. **麦克斯韦-玻尔兹曼分布**：描述了理想气体中粒子的速度分布，是统计力学的经典结果，解释了气体的温度与粒子平均动能之间的关系。 5. **集合论**：在统计力学中，集合论提供了处理大量粒子系统的数学工具，用于定义和操作各种状态空间。 6. **布朗运动**：统计力学中的一个重要应用是布朗运动理论，它描述了微小颗粒在流体中的随机运动，是分子热运动的宏观表现，也是对分子运动论的实验证据。 7. **相变和临界现象**：统计力学可以解释物质在不同相（如固态、液态和气态）之间转换的过程，以及临界点附近的行为，如磁性材料的居里点或液体的沸点。 8. **路径积分**和**量子统计**：统计力学也包含了量子力学的概念，如路径积分方法，用于处理量子系统的统计行为。 9. **系综理论**：系综理论提供了一种框架，通过考虑不同的统计权重来计算宏观量，例如正则系综、微正则系综和大数系综。 10. **涨落理论**：统计力学探讨了宏观量的波动，即涨落，这在理解和预测非平衡态过程时至关重要。本书不仅涵盖了这些基础概念，还可能涉及更现代的主题，如超导性、超流动性、自组织现象、复杂系统和混沌理论等。它对理解自然界中广泛存在的物理现象，以及开发新的理论和实验技术，都具有深远意义。

Classical Statistical Mechanics

=

if 

is the cell containing (

q

). The evolution (1.2.3) may send

a few particles outside the volume

, cubic for simplicity,thatwe imagine to

contain the particles: one has therefore to supplement (1.2.3) by

boundary

conditions

that will tell us the physical nature of the walls of

They could be

reecting

, if the collisions with the walls are elastic, or

periodic

if the opp osite faces of the region

are identied (a very convenient

\mathematical ction", useful to test various models and to minimize the

\nite size" eects,

i.e.

dependence of observations on system size).

One cannot, however, escap e some questions of principle on the structure

of the map

that it is convenient not to ignore, although their deep under-

standing may become a necessity only on a second reading.

First we shall neglect the p ossibilitythat(

q

) is on the boundary of a cell

(a case in which

is not uniquely determined, but whichcanbeavoided

by imagining that the cells walls are slightly deformed).

More imp ortant, in fact crucial, is the question of whether



implies 

= 

: the latter is a property which is certainly true in the

case of point cells (

= 0), b ecause of the uniqueness of the solutions of

dierential equations. It has an obvious intuitive meaning and an interest

due to its relation with reversibilityof motion.

In the following analysis a key role is played by Liouville's theorem which

tells us that the transformation mapping a generic initial datum (

pq

)into

the conguration (

q

(

pq

)isa

volume preserving

transformation.

This means that the set of initial data (

pq

)in evolves in the time



into

aset

 with volume equal to that of . Although having the same volume

of 

it wil l no longer have the same form

of a square parallelepip ed with

dimensions

p

q

. For

small it will b e a rather small parallelepip ed ob-

tained from  via a linear transformation that expands in certain directions

while contracting in others.

It is also clear that in order that the representation of the microscopic

states of the system be consistent it is necessary to imp ose some non trivial

conditions on the time interval, so far unsp ecied, that elapses between

successive (thought) observations of the motions. Such conditions can be

understoo d via the following reasoning.

Suppose that

is very small (actually by this wemean,here and below,

that

both

p

and

q

are small) so that the region

 can be regarded as

obtained by translating  and possibly by deforming it via a linear dilata-

tion in some directions or a linear contraction in others (contraction and

dilatation

balance

each other because, as remarked, the volume remains

constant). This is easily realized if

is small enough since the solutions to

ordinary dierential equations can always be thoughtof,lo cally, as linear

transformations close to the identity, for small evolution times



. Then:

i) If

dilates and contracts in various directions, even by a small amount,

there must necessarily exist pairs of distinct cells 

=

for which



: an example is provided by the map of the plane transforming (

x y

)

into

(

x y

)=((1+

)

;

x

(1 +

)

">

0 and its action on the lattice of

Classical Statistical Mechanics

the integers. Assuming that one decides that the cell 

into which a given

cell  evolves is, among those whichintersect its image

, the one which

had the largest intersection with it, then indetermination arises for a set of

cells spaced byabout

;

It is therefore necessary that



be small, say:

<#

with

such that the map

(associated with



, see (1.2.2), hence close

to the identity) produces contractions and expansions of  that can be

neglected for the

large majority

of the cells . Only in this way will it b e

possible that



with 

=

for just a small fraction of the cells

and, hence, one can hope that this p ossibility is negligible.

It should b e remarked explicitly that the abovepoint of view is systemat-

ically taken byphysicists p erforming numerical exp eriments.

Phase space

is representedincomputers as a nite

,butvery large, set of points whose

positions are changed by the time evolution (how many dep ends on the

precision of the representation of the reals). Even if the system studied is

modeled by a nice dierential equation with global uniqueness and existence

of solutions, the computer program, while trying to generate a permutation

of phase space points, will commit errors,

i.e.

two distinct points will be

sent to the same p oint(we do not talk here of round-o errors, whichare

not really errors as they are

a priori

known, in principle): one thus hop es

that such errors are rare enough to be negligible. This seems inevitable

except in some remarkable cases, the only nontrivial one I know of being in

LV93].

ii)

But



cannot be too small either, if one wishes to maintain coherently

the point of view that microscopic states are describ ed by phase space cells.

In fact to a cell  is asso ciated a natural time scale

;

(): which can be

dened as the minimum time in order that  becomes distinguishable from

the cell into which it evolves in time

;

(). And



must be necessarily

larger than the latter

minimum

time scale:

;

()

<

(otherwise wehave Zeno's paradox and nothing moves).

Summarizing we can say that in order to be able to dene the dynamic

evolution as a map p ermuting the phase space cells it must be that



chosen so that:

;

max



;

()







where the quotes mean that the maximum has to be taken as  varies

within the \ma jority" of the cells, where one can suppose that 

= 

implies



Classical Statistical Mechanics

One should realize that if

is a \reasonable" molecular p otential (a typical

model for

is, for instance, the

Lennard-Jones potential

with

intensity

and

range

given by:

(

)=4

((

)

;

(

)

)) it will generically b e that:

lim

;

=0 (1

while for small

the right hand side of (1.2.6) (which has a purely kine-

matical nature) b ecomes

independent.

Hence it will b e p ossible, at least in the limit in which

tendsto0,todene



so that (1.2.4), (1.2.5) hold

i.e.

it will be p ossible to fulll the above

consistency criteria for the describing microscopic states of the system via

nite cells.

On the other hand if

0, and

a posteriori

one should think that



;

erg sec

, the question we are discussing becomes quite delicate:

were it not b ecause we do not know what we should understand when we

say

the \large majority"

of the phase space cells.

In fact, on the basis of the results of the theory it will become possible to

evaluate the in$uence on the results themselves of the existence of pairs of

cells 

=

with



Logically at this point the analysis of the question should be postponed

until the consequences of the hyp otheses that we are assuming allowusto

reexamine it. It is nevertheless useful, in order to better grasp the delicate

nature of the problem and the orders of magnitude involved, to anticipate

some of the basic results and to provide estimates of



': readers preferring

to think in purely classical terms, by imagining that

= 0 on the basis of

a dogmatic interpretation of the (classical) atomic hyp othesis, can skip the

discussion and proceed by systematically taking the limit as

0 of the

theory that follows.

It is however worth stressing that setting

= 0 is an illusory simplication

avoiding posing a problem that is to daywell known to be deep. Assuming

that, at least in principle, it should b e possible to measure exactly p ositions

and momenta of a very large number of molecules (or even of a single one)

means supp osing it is possible to perform a physical op eration that

no one

would b e able to perform. It was the obvious di%culty, one should recall, of

such an operation that in the last century made it hard for some to accept

the atomic hypothesis.

Coming to the problem of providing an idea of the orders of magnitude of



one can interpret \max" in (1.2.6) as evaluated by considering as

typical

cells those for which the momenta and the recipro cal distances of the parti-

cles takevalues \close" to their \average values". The theory of

statistical

ensembles

(see below) will lead to a natural probability distribution giving

the probability of each cell in phase space, when the system is in macro-

scopic equilibrium. Therefore we shall be able to compute, by using this

probability distribution, the average values of various quantities in terms of

macroscopic quantities like the absolute temp erature

, the particle mass

, the particle number

, and the volume

available to the system.

Classical Statistical Mechanics

The main property of the probability distributions of the microscopic states

observed in a situation in which the macroscopic state of the system is in

equilibrium is that the average velocity and average momentum

v p

will b e

related to the temperature by:

T mv

where



;

erg



;

is the universal Boltzmann's constant.

Other relevantquantities are the characteristic parameters of the interac-

tion,

i.e.

the

strength

with the dimension of energy and the

range

with

the dimension of length. It follows from the developments of the theory

of equilibrium statistical mechanics,

independently of the particular form of

(

) (as long as it is \reasonable", like for instance the abovementioned

Lennard-Jones potential), that



where

is the critical liquefac-

tion temp erature and

is of the order of the molecular diameter (between



;

and 4



;

in the simplest gases like

HeO

CO

,see

Chap.V).

We estimate

rst (the time scale over which expansion and contraction

of a phase space cell become sensible) lo oking at a typical cell where one

can assume that the particles evolveintime without undergoing multiple

collisions. In such a situation the relativevariation of a linear dimension of

 in the time



will be, for small



, prop ortional to



and it may depend on

" m r



: the pure numbers related to



and to the phase space dilatations

(

i.e.

to the derivatives of the forces appearing in the equations of motion)

that one can form with the above quantities are



(

)

and



(

)

Hence the phase space changes in volume will be negligible, recalling that

, from (1.2.8), and setting



,provided:

<

min

















The condition



"=mr

1 means that, even during a microscopic collision

taking place while the time



elapses, there is no sensible expansion while

the second condition

 <r

= v

means that the time



is short with respect

to the total collision duration (which therefore takes several units of



be completed).

To estimate

;

(the time scale over which a cell evolves enough to be

distinguishable from itself ) note that, given , the co ordinates



q



of the

phase space p oints in the cell  change obeying the Hamiltonian equations

of motion, in the time



,by:

q









(

q

)





(1)



p









(

q

)





(2)



q

10)

Classical Statistical Mechanics

where



(1)





(2)



are the variations of the energy

in the cell  when the

coordinates



or, resp ectively,



vary by the amount

p

q

i.e.

they

vary by a quantity equaling the linear dimensions of the cell , while the

others stay constant (so that the variations in (1.2.10) are related to the

partial derivatives of the energy function

Dening therefore the

energy indetermination

,whichwe denote by

E

(),

in the cell  as:

E

() = max



(



(1)



E 

(2)



) (1

11)

we see that the minimum time

;

() that one has to wait in order to see

that the cell evolves into a cell whichis

distinguishable

from  itself is:

;

() max





(1)



p



q

;

() max





(2)



q



p

12)

in fact,

p

and

q

being the linear dimensions of , Eq. (1.2.12) just says

that at least one of the sides of  has moved away by a quantity of the

order of its length (thus becoming distinguishable from itself ).

Since we set

p q

one deduces from (1.2.11),(1.2.12):

;

()

E

()



13)

and

;



t

can be chosen so that, introducing the notation:

E

min



E

() (1

14)

wehave:

tE

15)

We can therefore see, on the basis of (1.2.9),(1.2.15), whether or not an

interval (

;

#

) admissible for



exists. We can in fact imagine that

p

hence

E

p p=m



and:

;



h=k

16)

Equation (1.2.16) gives a remarkable interpretation of the time scale

h=k

it is the time necessary so that a phase space cell, typical among those

describing the microscopic equilibrium states at temp erature

, becomes

distinguishable from itself.

One can say, dierently,that

;

is determined by the size of _

p

i.e.

the size of the rst derivatives of the Hamiltonian, while

is related to the

phase space expansion,

i.e.

to the second derivatives of the Hamiltonian.

With some algebra one derives, from (1.2.9), (1.2.16):

;

)

min(

T=T



(

T=T

)

17)

Therefore it is clear that the relation

;

necessary for a consistent

description

of the microscopic states in terms of phase space cells, will b e

剩余358页未读，继续阅读

点点吃得太多了

粉丝: 180
资源: 683

统计力学精要：加拉沃蒂的短篇论述

Pathria Statistical Mechanics.djvu

Information Theory and Statistical Mechanics_Jaynes

Information Theory and Statistical Mechanics

Nonequilibrium statistical mechanics

Nonequilibrium Statistical Mechanics

Statistical Mechanics of Learning

Statistical mechanics of nonequilibrium liquids

Statistical Mechanics of Complex Networks

Statistical Mechanics of complex networks

Greiner - Thermodynamics and statistical mechanics

最新资源