Lectures TSP - Theo III (WiSe 19/20) in Englisch, ausformuliert PDF

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Lectures on ThermodynamicsandStatistical PhysicsGleb Arutyunova∗aII. Institut f ̈ur Theoretische Physik, Universit ̈at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Zentrum f ̈ur Mathematische Physik, Universit ̈at Hamburg, Bundesstrasse 55, 20146 Hamburg, GermanyAbstractThis course covers t...

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Lectures on Thermodynamics and Statistical Physics Gleb Arutyunova∗ a

II. Institut f¨ ur Theoretische Physik, Universit¨ at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany Zentrum f¨ ur Mathematische Physik, Universit¨ at Hamburg, Bundesstrasse 55, 20146 Hamburg, Germany

Abstract This course covers the basic concepts of equilibrium thermodynamics and statistical physics. Staring from foundations, we extensively discuss the laws of thermodynamics including their modern and historical formulations. We introduce the method of thermodynamic potentials and use it to establish the conditions of thermodynamic equilibrium. Concerning statistical physics, we start with introducing the main statistical distributions (Bose-Einstein, FermiDirac and Maxwell-Boltzmann) for an ideal gas by means of a simplified method of boxes and cells and later describe Gibbs’ general method of statistical ensembles.

Last update: 15.10.2019

∗

Email: [email protected]

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Contents I

Equilibrium thermodynamics

5

1 Foundations

6

1.1

Thermodynamic systems – what they are . . . . . . . . . . . . . . . . . . .

6

1.2

Choice of state parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.3

Equations of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.4

Quasi-stationary, reversible and irreversible processes . . . . . . . . . . . . .

16

1.5

Examples of thermodynamic systems . . . . . . . . . . . . . . . . . . . . . .

16

2 Laws of thermodynamics

20

2.1

The first law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.2

The second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

2.3

Alternative formulations of the second law . . . . . . . . . . . . . . . . . . .

30

2.4

Absolute temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35

2.5

The second law for non-quasi-static processes . . . . . . . . . . . . . . . . .

38

2.6

The third law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Thermodynamic potentials

II

38 41

3.1

Main thermodynamic potentials . . . . . . . . . . . . . . . . . . . . . . . . .

3.2

Conditions of thermodynamics equilibrium . . . . . . . . . . . . . . . . . . .

48

3.3

Thermodynamics of chemical reactions . . . . . . . . . . . . . . . . . . . . .

48

3.3.1

Condition of chemical equilibrium . . . . . . . . . . . . . . . . . . .

48

3.3.2

The law of mass action

. . . . . . . . . . . . . . . . . . . . . . . . .

Statistical physics

41

49

51

1 Foundations

52

2 Statistical distributions for ideal gases

53

2

2.1

Boxes and cells in the phase space . . . . . . . . . . . . . . . . . . . . . . .

2.2

Bose-Einstein and Fermi-Dirac distributions . . . . . . . . . . . . . . . . . .

2.3

Boltzmann principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

2.4

Maxwell-Boltzmann distribution . . . . . . . . . . . . . . . . . . . . . . . .

64

2.5

Classical ideal gas. Degeneration criterion. . . . . . . . . . . . . . . . . . . .

68

2.6

Grand canonical potential for bose and fermi gases . . . . . . . . . . . . . .

71

2.7

Quantisation of energy and Nernst theorem . . . . . . . . . . . . . . . . . .

72

3 Maxwell-Boltzmann gas

55

76

3.1

Monoatomic classical Maxwell-Boltzmann gas . . . . . . . . . . . . . . . . .

76

3.2

Maxwell distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

77

3.3

Classical multi-atomic gases . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4 Gibbs’ method 4.1

4.2

III

53

81

Liouville theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.1.1

Γ-space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.1.2

Ensembles and Liouville theorem . . . . . . . . . . . . . . . . . . . .

82

Microcanonical distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

Mathematical appendix

89

1. The volume of N -dimensional sphere . . . . . . . . . . . . . . . . . . . . . . .

90

2. Stirling formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

3. Gauss distribution for one and two variables . . . . . . . . . . . . . . . . . . .

3

92

General references [1] Herbert Callen, Thermodynamics and Introduction to Thermostatistics, Jonh Wiley & Sons, Inc., 1985. [2] Walter Greiner, Ludwig Neise, Horst St¨ocker, Thermodynamics and Statistical Mechanics, Springer, 1994. [3] Lev Landau and Evgeny Lifshitz, Statistical Physics: Volume 5 (Course of Theoretical Physics, Volume 5), Elsevier, 3d edition, 1980. [4] Franz Schwabl, Statistische Mechanik, Springer-Verlag Berlin Heidelberg New York, 2000. [5] Hermann Schulz, Statistische Physik, Wissenschaftlicher Verlag Harri Deutsch GmbH, Frankfurt am Mein, 2005. [6] Jochen Rau, Statistical Physics and thermodynamics, Oxford, 2017.

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Organization of the course The lectures are scheduled on Mondays 8.30-10.00 and Thursdays 8.30-10.00 at H¨ orsaal I (Wolfgang Pauli-H¨ orsaal), Jungius street 9. The first lecture is on 14.10.2019. The schedule of classes is 1. Group A (german speaking, Philipp Englert): Monday, 10.15-11.45, Seminar room 4 2. Group B (english speaking, Dr. Sylvain Lacroix): Monday, 10.15-11.45, Seminar room 6 3. Group C (english speaking, Dr. Sylvain Lacroix): Monday, 12.00-13.30, Seminar room 4 4. Group D (english speaking, Alejandro Romero Ros): Monday, 12.00-13.30, Seminar room 6 5. Group E (english speaking, Cristian Bassi): Monday, 14.30-16.00, Seminar room 3 6. Group F (english speaking, Jie Chen): Monday, 14.30-16.00, Seminar room 4 7. Group G (english speaking, Simos Mistakidis): Monday, 14.30-16.00, Seminar room 6

The fist class is on 21.10.2019. In total there are 13 exercises classes. Tutors of the 7 groups above Victor Danescu - [email protected] (3 groups) Friethjof Theel - [email protected] (1 group) Hannes Kiehn - [email protected] (1 group) Nejira Pintul - [email protected] (1 group) Constantin Harder - constantin.harder.desy.de (1 group) The final exam is scheduled on 10.02.2020 (Monday), 10.00-12.00 in H¨orsaal I and H¨orsaal II. Retake will take place on 23.03.2020 (Monday), 11.00-13.00 in H¨orsaal I.

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Part I

Equilibrium thermodynamics

6

Chapter 1

Foundations This section lays the foundations of equilibrium thermodynamics. We introduce a notion of a thermodynamic system, discuss state parameters and equations of state.

1.1

Thermodynamic systems – what they are

Thermodynamics and statistical physics, as branches of physics, are applicable to the socalled thermodynamical (statistical) systems. These systems are specified by a set of their basic physical features rather than by a single property. These basic physical features are summarised below. 1. These are systems of many particles interacting with each other and with external fields . The word “particle” stands here for molecules, if the corresponding system is a gas or a liquid, and groups of atoms making sides of a crystal lattice in a solid body. “Many particles” means that the number of particles in a system is measured in units of Avogardo’s number N0 N0 = 6.022 . . . · 1023. In other words, in thermodynamics both the amount of matter N , i.e. the number of structural units, and its change dN are measured by a number of moles n = N/N0 . Thus, thermodynamic systems are the systems of laboratory size. From the point of view of classical mechanics, an analytic description of motion of such multi-body systems is hopeless. In practice, since N ∼ N0 ≫ 1, in performing calculations with thermodynamic systems one can neglect the terms of order O(1/N ) in comparison to 1.

Since for thermodynamic systems N is so large that N/N0 is visibly finite, such systems are macroscopic and measurements of their physical parameters are naturally performed with the help of macroscopic devices (thermometers, manometers and so on). This means that a device simultaneously interacts with a macroscopically large number of particles and measures an average value of the corresponding physical quantity. 2. For any thermodynamic system there exists a state of thermodynamic equilibrium that the system spontaneously reaches after some time under fixed

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2 2 3 3 Figure 1.1: Transitivity of a state of thermodynamic equilibrium. external conditions . This statement constitutes the so-called zeroth law of thermodynamics. A state of thermodynamic equilibrium is the state where all macroscopic parameters of the system do not change with time and where macroscopic flows of any kind are absent.1 The property of reaching the state of thermodynamic equilibrium is not characteristic to mechanical systems, especially in view of the the Poincar´e recurrence theorem (H. Poincar´e, 1890). This theorem states that certain systems will, after a sufficiently long but finite time, return to a state very close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state. Estimates show, however, that this finite time interval includes for one mole of a substance a factor of order 10N0 , which is tremendously huge in comparison to the age of the universe (1017 − 1018 s). The state of thermodynamic equilibrium has two important properties:

• it is a dynamical state. From a microscopic point of view the parameters of such a state are not strictly fixed in time but fluctuate in time. Also the fluxes of the number of particles, energy, etc fluctuate around their vanishing mean value. One can show that fluctuations δF = F (N , t) − F¯ of a quantity F around its mean value F¯ scale with N as δF ∼ N −1/2 . F¯ It is then clear that for N ≫ 1 spontaneous deviations from an equilibrium state are highly suppressed and the zero’th postulate of thermodynamics practically holds true. • A thermodynamic state exhibits a property of thermodynamic transitivity. This property means that if a thermodynamic system 1 in a state of thermodynamic equilibrium 1

We emphasise that an equilibrium state is not the same as a stationary state. In a stationary state the macroscopic state quantities are also time-independent, but these states are always connected with the energy flux, which is not the case for equilibrium states. For instance, let us consider an electric hot plate of a usual household. If one puts a pot with a meal on top of it, after some time a stationary state will be attained where the temperature of the meal will not change any longer. This, however, is not a state of thermal equilibrium as long as the surroundings have a different temperature. One must continuously supply the system with the (electrical) energy to prevent the cooling of the dish, which continuously radiates energy (heat) into the surroundings. This system is not isolated since the energy is supplied as well as emitted.

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being successively in a thermal contact with equilibrium systems 2 and 3 does not change its state, then bringing in a thermal contact systems 2 and 3 does not change their equilibrium states. The property of thermodynamic transitivity is in the heart of the notion of temperature as a new non-mechanical characteristic of an equilibrium state. The numerical value of this characteristics can be determined by the value of some mechanical parameter of system 1, which for some reasons is chosen to play a role of a thermometer. 3. Thermodynamic systems obey the additivity principle with respect to the amount of substance (the number of particles N ) or its volume V – all parameters characterising an equilibrium state can only belong to one of the two additivity classes, namely, to the class 0, or 1 . These parameters do not depend on systems with which a given system is in contact, in particular, which boundary conditions are fixed (type of walls of a container). According to the additivity principle, all quantities that describe a thermodynamic system belong to one of the two additivity classes depending on how the value of the corresponding quantity reacts on the devision of an equilibrium thermodynamic system on two equilibrium macroscopic parts: • if a value of a thermodynamic quantity under its devision on two macroscopic parts 1 and 2 behaves as F1+2 = F1 + F2 , then this quantity is called additive or extensive (or the 1-st additivity class), • if under such a devision the value of this quantity remains unchanged and is the same for each of its parts, f1+2 = f1 = f2 , then this quantity is called non-additive (or the 0-th additivity class). Examples of additive quantities constitute the number N of particles in a system, its volume V , total energy E , heat capacity C, etc. To non-additive quantities belong temperature of a system θ, its pressure p, the chemical potential µ and specific quantities of additive characteristics, like specific energy ε = E /N , specific heat capacity c = C/N , etc. In the framework of quasi-static and equilibrium thermodynamics macroscopic characteristics of other additivity classes than 0 and 1 simply do not exist. 4. To thermodynamic systems the I, II and III laws of thermodynamics are applicable . In fact these are axioms (or postulates) which are traditionally called the laws of thermodynamics and their applicability to a system serves as an indication of its thermodynamic nature.

1.2

Choice of state parameters

A choice of the description of a thermodynamic system is the same as a choice of some definite set of thermodynamic state parameters which unambiguously characterise an equilibrium state. In fact, this set of parameters is distinguished by a procedure (an experiment) 9

by means of which we single out the system from a surrounding media by putting some physical or imaginary walls. The procedure is not unique and essentially depends on concrete problems one wants to solve. Let us stress from the very beginning that the thermodynamic substance which we single out from surroundings remains the one and the same and therefore it keeps its properties regardless of the procedure used. The most important choices of singling out the system from surroundings are the following. α) Adiabatically isolated system. This is a system which is singled out by means of adiabatic walls α which are impenetrable to heat, so that there is no heat exchange with surroundE , V, a, N ings. Singling out a system in such a way fixes 2 its volume, number of particles N , an external field a penetrating through an adiabatic wall and the energy of E of the system which is the energy of all particles inside the adiabatic shell. All the parameters fixed by adiabatic walls are mechani- Figure 1.2: Adiabatically isolated syscal, among them there is no any specific thermo- tem. All parameters are mechanical. dynamic parameter. An adiabatically isolated system can exchange energy with surroundings but only in the form of work, not in the form of heat or matter.3 Graphically, we single out an adiabatically isolated system by double walls like the walls of a dewar vessel, see Fig. 1.2. β) System in a thermostat (heat bath). A system is singled out by means of thermallyconducting walls β. In this description two systems participate – the first, which is the one we are interested in, and the second, which is in the thermal equilibrium with the first and which serves as the thermometer. This second, external system is called thermostat or heat bath.

System

θT

θ, V, a, N

Thermostat

A thermostat can be large or small, what matters is that it is all the time at a constant temperature θT that coincides with the temperFigure 1.3: A system in a thermostat. ature θ of the system we are interested in. Thus, this way of singling out a system fixes the following parameters θ, V, a, N . Because of walls transparent to heat, energy of the system in the precise mechanical sense is not fixed, there is a constant exchange of energy with the thermostat through the walls. γ) System singled out by imaginary walls. We consider a big thermodynamic system in equilibrium and single out by a mental effort some part of it which, will be an object 2

If there are particles of different sorts, then the number of particles of each sort Ni are fixed. All thermodynamic systems can be split into three general classes: open, closed and isolated. Open systems can exchange with surroundings both matter and energy. Closed systems can exchange only energy but not matter. Finally, isolates systems do not interact in any way with surroundings. For instance, the case α is a closed system, as it can exchange energy with surroundings in the form of mechanical work. 3

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of our investigation. Thus, we fix volume V (which is a geometric factor), temperature θ , which coincides with the temperature of the rest of the system θT and the field(s) a. The exact number of particles N is not fixed. Because if this, one might get a feeling that a number of parameters needed for a description of a thermodynamic state in the present situation get reduced in comparison to α and β. This is, however, not so. Instead of N one can choose System another parameter µ (in the case of a multiθ, V, a, µ θT component system, µ = {µi }). This parameter is called chemical potential and its meaning will be clarified in due course. Thermostat δ) System under a piston. A system is singled out from a thermostat by thermallyconductive walls, as in the case β, but one Figure 1.4: Singling out a system by of the walls is movable, so that pressure in imaginary walls. the thermostat pT is transferred to the system. Thus, with respect to the system under study the thermostat plays not only the role of a thermometer, θT = θ, but also the role of a manometer measuring pressure pT = p. In the present description of the system the parameters θ, p, a, N are fixed. In particular, volume V is not needed to be fixed as it will adjust itself automatically to a certain value by a movable wall. One can imagine other ways of of singling out a thermodynamic system from surroundings but the four ways discussed above are enough pT for our further purposes. We see that a choice of thermodynamic parameters to describe a therθT θ, p, a, N System modynamic state corresponds to the conditions of an experiment, which consists in setting up different boundary conditions for a thermodyThermostat namic system under study. In principle, the one and the same system put in a vessel with different conditions on the boundary might behave Figure 1.5: A system under a piston. itself differently. However, the specifics of thermodynamics approach to t...