Basic Theoretical Physics: A Concise Overview P30 docx

Part IV Thermodynamics and Statistical Physics 39 Introduction and Overview to Part IV This is the last course in our compendium on theoretical physics. In Ther- modynamics and Statistical Physics we shall make use of a) classical non- relativistic mechanics as well as b) non-relativistic quantum mechanics and c) aspects of special relativity. Whereas in the three above-mentioned sub- jects one normally deals with just a few degrees of freedom (i.e. the number of atoms N in the system is usually 1 or of the order of magnitude of 1), in thermodynamics and statistical physics N is typically ≈ 10 23 ; i.e. the num- ber of atoms, and hence degrees of freedom, in a volume of ≈ 1cm 3 of a gas or liquid under normal conditions is extremely large. Microscopic properties are however mostly unimportant with regard to the collective behavior of the system, and for a gas or liquid only a few macroscopic properties, such as pressure p, temperature T and density  characterize the behavior. Quantum mechanics usually also deals with a small number of degrees of freedom, however with operator properties which lead to the possibility of discrete energy levels. In addition the Pauli principle becomes very important as soon as we are dealing with a large number of identical particles (see below). In classical mechanics and non-relativistic quantum mechanics we have v 2  c 2 , with typical atomic velocities of the order of |ψ|ˆvψ| ≈ c 100 . However statistical physics also includes the behavior of a photon gas, for example, with particles of speed c.Hereofcoursespecial relativity has to be takenintoaccount(seePartI) 1 . In that case too the relevant macroscopic degrees of freedom can be de- scribed by a finite number of thermodynamic potentials, e.g., for a photon gas by the internal energy U(T,V,N)andentropy S(T,V,N), or by a single combination of both quantities, the Helmholtz free energy F (T,V,N):=U (T,V,N) −T ·S(T,V,N) , where T is the thermodynamic temperature of the system in degrees Kelvin (K), V the volume and N the number of particles (number of atoms or 1 In some later chapters even aspects of general relativity come into play. 302 39 Introduction and Overview to Part IV molecules). In phenomenological thermodynamics these thermodynamic po- tentials are subject to measurement and analysis (e.g., using differential cal- culus), which is basically how a chemist deals with these quantities. A physicist however is more likely to adopt the corresponding laws of statistical physics, which inter alia make predictions about how the above functions have to be calculated, e.g., U(T,V,N)=  i E i (V,N)p i (T,V,N) . Here {E i ≡ E i (V,N)} is the energy spectrum of the system, which we assume is countable, and where the p i are thermodynamic probabilities, usually so- called Boltzmann probabilities p i (T, )= exp − E i k B T  j exp − E j k B T , with the Boltzmann constant k B =1.38 10 −23 J/K. We note here that a temperature T ≈ 10 4 K corresponds to an energy k B T of ≈ 1eV. Using a further quantity Z(T,V,N), the so-called partition function, where Z(T,V,N):=  j exp −βE j and β := 1 k B T , we obtain U(T,V,N) ≡− d dβ ln Z ; F (T,V,N) ≡−k B T · ln Z ; S(T,V,N) ≡− ∂F ∂T ≡−k B  j p j ln p j . There are thus three specific ways of expressing the entropy S: a) S = U − F T , b) S = − ∂F ∂T , and c) S = −k B  j p j ln p j . The forms a) and b) are used in chemistry, while in theoretical physics c) is the usual form. Furthermore entropy plays an important part in information theory (the Shannon entropy), as we shall see below. 40 Phenomenological Thermodynamics: Temperature and Heat 40.1 Temperature We can subjectively understand what warmer or colder mean, but it is not easy to make a quantitative, experimentally verifiable equation out of the inequality T 1 >T 2 . For this purpose one requires a thermometer, e.g., a mer- cury or gas thermometer (see below), and fixed points for a temperature scale, e.g., to set the melting point of ice at normal pressure ( ˆ= 760 mm mercury column) exactly to be 0 ◦ Celsius and the boiling point of water at normal pressure exactly as 100 ◦ Celsius. Subdivision into equidistant inter- vals between these two fixed points leads only to minor errors of a classical thermometer, compared to a gas thermometer which is based on the equation for an ideal gas: p ·V = Nk B T where p is the pressure ( ˆ= force per unit area), V the volume (e.g., = (height) x (cross-sectional area)) of a gas enclosed in a cylinder of a given height with a given uniform cross-section, and T is the temperature in Kelvin (K), which is related to the temperature in Celsius, Θ,by: T = 273.15 + Θ (i.e., 0 ◦ Celsius corresponds to 273.15 Kelvin). Other temperature scales, such as, for example, Fahrenheit and R´eaumur, are not normally used in physics. As we shall see, the Kelvin temperature T plays a particular role. In the ideal gas equation, N is the number of molecules. The equation is also written replacing N · k B with n Mol · R 0 ,whereR 0 is the universal gas constant, n Mol the number of moles, n Mol := N L 0 , and R 0 = L 0 · k B . Experimentally it is known that chemical reactions occur in constant propor- tions (Avogadro’s law), so that it is sensible to define the quantity mole as a specific number of molecules, i.e., the Loschmidt number L 0 given by ≈ 6.062(±0.003) · 10 23 . 304 40 Phenomenological Thermodynamics: Temperature and Heat (Physical chemists tend to use the universal gas constant and the number of moles, writing pV = n Mol · R 0 T, whereas physicists generally prefer pV = N ·k B T .) In addition to ideal gases physicists deal with ideal paramagnets,which obey Weiss’s law, named after the French physicist Pierre Weiss from the Alsace who worked in Strasbourg before the First World War. Weiss’s law states that the magnetic moment m H of a paramagnetic sample depends on the external field H and the Kelvin temperature T in the following way: m H = C T H, where C is a constant, or H = m H C ·T. From measurements of H one can also construct a Kelvin thermometer using this ideal law. On the other hand an ideal ferromagnet obeys the so-called Curie-Weiss law H = m H C · (T − T c )whereT c is the critical temperature or Curie temperature of the ideal ferromagnet. (For T<T c the sample is spontaneously magnetized, i.e., the external magnetic field can be set to zero 1 .) Real gases are usually described by the so-called van der Waals equation of state p = − a v 2 + k B T v − b , which we shall return to later. In this equation v = V N , and a and b are positive constants. For a = b = 0 the equation reverts to the ideal gas law. A T,p-diagram for H 2 O (with T as abscissa and p as ordinate), which is not presented, since it can be found in most standard textbooks, would show three phases: solid (top left on the phase diagram), liquid (top right) and gaseous (bottom, from bottom left to top right). The solid-liquid phase boundary would be almost vertical with a very large negative slope. The negative slope is one of the anomalies of the H 2 O system 2 . One sees how 1 A precise definition is given below; i.e., it turns out that the way one arrives at zero matters, e.g., the sign. 2 At high pressures ice has at least twelve different phases. For more information, see [36]. 40.2 Heat 305 steep the boundary is in that it runs from the 0 ◦ C fixed point at 760 Torr and 273.15 K almost vertically downwards directly to the triple point,which is the meeting point of all three phase boundaries. The triple point lies at a slightly higher temperature, T triple = 273.16 K , but considerably lower pressure: p triple ≈ 5 Torr. The liquid-gas phase bound- ary starting at the triple point and running from “southwest” to “northeast” ends at the critical point, T c = 647 K, p c = 317 at. On approaching this point the density difference Δ :=  liquid −  gas decreases continuously to zero. “Rounding” the critical point one remains topologically in the same phase, because the liquid and gas phases differ only quantitatively but not qualitatively. They are both so-called fluid phases. The H 2 O system shows two anomalies that have enormous biological con- sequences. The first is the negative slope mentioned above. (Icebergs float on water.) We shall return to this in connection with the Clausius-Clapeyron equation. The second anomaly is that the greatest density of water occurs at 4 ◦ C, not 0 ◦ C. (Ice forms on the surface of a pond, whereas at the bottom of the pond the water has a temperature of 4 ◦ C.) 40.2 Heat Heat is produced by friction, combustion, chemical reactions and radioactive decay, amongst other things. The flow of heat from the Sun to the Earth amounts to approximately 2 cal/(cm 2 s) (for the unit cal: see below). Fric- tional heat (or “Joule heat”) also occurs in connection with electrical resis- tance by so-called Ohmic processes, dE = R · I 2 dt, where R is the Ohmic resistance and I the electrical current. Historically heat has been regarded as a substance in its own right with its own conservation law. The so-called heat capacity C V or C p was defined as the quotient ΔQ w ΔT , where ΔQ w is the heat received (at constant volume or constant pressure, respectively) and ΔT is the resulting temperature change. Similarly one may define the specific heat capacities c V and c p : c V := C V m and c p := C p m , 306 40 Phenomenological Thermodynamics: Temperature and Heat where m is the mass of the system usually given in grammes (i.e., the molar mass for chemists). Physicists prefer to use the corresponding heat capacities c (0) V and c (0) p per atom (or per molecule). The old-fashioned unit of heat “calorie”, cal, is defined (as in school physics books) by: 1 cal corresponds to the amount of heat required to heat 1 g of water at normal pressure from 14.5 ◦ C to 15.5 ◦ C. An equivalent defi- nition is: (c p ) |H 2 O;normal pressure,15 ◦ C ! =1cal/g. Only later did one come to realize that heat is only a specific form of energy, so that today the electro-mechanical equivalent of heat is defined by the following equation: 1cal = 4.186 J = 4.186 Ws . (40.1) 40.3 Thermal Equilibrium and Diffusion of Heat If two blocks of material at different temperatures are placed in contact, then an equalization of temperature will take place, T j → T ∞ ,forj =1, 2, by a process of heat flowing from the hotter to the cooler body. If both blocks are insulated from the outside world, then the heat content of the system, Q w|“1+2” , is conserved, i.e., ΔQ w|“1+2” = C 1 ΔT 1 + C 2 ΔT 2 ≡ (C 1 + C 2 ) ·ΔT ∞ , giving ΔT ∞ ≡ C 1 ΔT 1 + C 2 ΔT 2 C 1 + C 2 . We may generalize this by firstly defining the heat flux density j w ,which is a vector of physical dimension [cal/(cm 2 s)], and assume that j w = λ · gradT (r,t) . (40.2) This equation is usually referred to as Fick’s first law of heat diffusion. The parameter λ is the specific heat conductivity. Secondly let us define the heat density  w . This is equal to the mass density of the material  M multiplied by the specific heat c p and the local temperature T (r,t)attimet, and is analogous to the electrical charge density  e . For the heat content of a volume ΔV one therefore has Q w (ΔV )=  ΔV d 3 r w . 3 3 In this part, in contrast to Part II, we no longer use more than one integral sign for integrals in two or three dimensions. 40.4 Solutions of the Diffusion Equation 307 Since no heat has been added or removed, a conservation law applies to the total amount of heat Q w (R 3 ). Analogously to electrodynamics, where from the conservation law for total electric charge a continuity law results, viz: ∂ e ∂t +divj e ≡ 0 , we have here: ∂ w ∂t +divj w ≡ 0 . If one now inserts (40.2) into the continuity equation, one obtains with div grad ≡∇ 2 = ∂ ∂x 2 + ∂ ∂y 2 + ∂ ∂z 2 the heat diffusion equation ∂T ∂t = D w ∇ 2 T, (40.3) where the heat diffusion constant, D w := λ  M c p , has the same physical dimension as all diffusion constants, [D W ]=[cm 2 /s]. (40.3) is usually referred to as Fick’s second law. 40.4 Solutions of the Diffusion Equation The diffusion equation is a prime example of a parabolic partial differential equation 4 . A first standard task arises from, (i), the initial value or Cauchy problem. Here the temperature variation T (r,t = t 0 ) is given over all space, ∀r ∈ G, but only for a single time, t = t 0 .RequiredisT (r,t) for all t ≥ t 0 . A second standard task, (ii), arises from the boundary value problem. Now T (r,t) is given for all t, but only at the boundary of G, i.e., for r ∈ ∂G.RequiredisT (r,t)overallspaceG. For these problems one may show that there is essentially just a single solution. For example, if one calls the difference between two solutions u(r,t), i.e., u(r,t):=T 1 (r,t) −T 2 (r,t) , 4 There is also a formal similarity with quantum mechanics (see Part III). If in (40.3) the time t is multiplied by i/ and D w and T are replaced by  2 /(2m)and ψ, respectively, one obtains the Schr¨odinger equation of a free particle of mass m.Here,iandψ have their usual meaning. 308 40 Phenomenological Thermodynamics: Temperature and Heat then because of the linearity of the problem it follows from (40.3) that ∂u ∂t = D w ∇ 2 u, and by multiplication of this differential equation with u(r,t) and subsequent integration we obtain I(t):= d 2dt  G d 3 ru 2 (r,t)=D w ·  G d 3 ru(r,t) ·∇ 2 u(r,t) = −D w  G d 3 r{∇u(r,t)} 2 + D w  ∂G d 2 Su(r,t)(n ·∇)u(r,t) . To obtain the last equality we have used Green’s integral theorem, which is a variant on Gauss’s integral theorem. In case (ii), where the values of T (r,t) are always prescribed only on ∂G, the surface integral ∝  ∂G is equal to zero, i.e., I(t) ≤ 0, so that  G d 3 r · u 2 decreases until finally u ≡ 0. Since we also have ∂ ∂t ≡ 0 , one arrives at a problem, which has an analogy in electrostatics. In case (i), where at t 0 the temperature is fixed everywhere in G, initially one has uniqueness: 0=u(r,t 0 )=I(t 0 ); thus u(r,t) ≡ 0 for all t ≥ t 0 . One often obtains solutions by using either 1) Fourier methods 5 or 2) so-called Green’s functions. We shall now treat both cases using examples of one-dimensional standard problems: 5 It is no coincidence that L. Fourier’s methods were developed in his tract “Th´eorie de la Chaleur”. 40.4 Solutions of the Diffusion Equation 309 1a)(Equilibration of the temperature for a periodic profile): At time t = t 0 =0 assume there is a spatially periodic variation in temperature T (x, 0) = T ∞ + ΔT (x) , with ΔT (x + a)=ΔT (x) for all x ∈ G. One may describe this by a Fourier series ΔT (x)= ∞  n=−∞ b n exp (ink 0 x) , with b n ≡ 1 a a  0 dx exp (−ink 0 x)ΔT (x) . Here, k 0 := 2π λ 0 , where λ 0 is the fundamental wavelength of the temperature profile. One can easily show that the solution to this problem is T (x, t)=T ∞ + ∞  n=−∞,=0 b n e ink 0 x ·e −n 2 t τ 0 , where 1 τ 0 = D w k 2 0 . According to this expression the characteristic diffusion time is related to the fundamental wavelength. The time dependence is one of exponential decay, where the upper harmonics, n = ±2, ±3, , are attenuated much more quickly, ∝ e −n 2 t τ 0 , than the fundamental frequency n = ±1. 1b)A good example of a “boundary value problem” is the so-called permafrost problem, which shall now be treated with the help of Fourier methods. At the Earth’s surface z = 0 at a particular location in Siberia we assume there is explicitly the following temperature profile: T (z =0,t)=T ∞ + b 1 cos ω 1 t + b 2 cos ω 2 t. T ∞ is the average temperature at the surface during the year, ω 1 = 2π 365d is the annual period and ω 2 = 2π 1d is the period of daily temperature fluctuations (i.e., the second term on the r.h.s. describes the seasonal variation of the daytime temperature average, averaged over the 24 hours of a day). We write cos ω 1 t = Ree −iω 1 t . we shall return to later. In this equation v = V N , and a and b are positive constants. For a = b = 0 the equation reverts to the ideal gas law. A T,p-diagram for H 2 O (with T as abscissa and. Equation The diffusion equation is a prime example of a parabolic partial differential equation 4 . A first standard task arises from, (i), the initial value or Cauchy problem. Here the temperature. Physics we shall make use of a) classical non- relativistic mechanics as well as b) non-relativistic quantum mechanics and c) aspects of special relativity. Whereas in the three above-mentioned