Statistical physics (2005)

larger than, or equal to, that of any thermal emitter at the same temperature, the black body is the best thermal emitter. This is the Kirchho law.

In the case of a di using body (reemitting radiation in a direction other than −θ) or of a fluorescent body (reemitting at a frequency di erent from that of its excitation), one can no longer express a detailed balance, angle by angle or frequency by frequency, between the ambient radiation and the body radiation. However, in steady-state regime, the total absorbed power is equal to the emitted power and the body temperature remains constant. The applicable relation is deduced from (9.59) after integration over the angles and/or over the frequencies.

One can also say that the power, assumed to be emitted by a black body, received by the studied body at thermal equilibrium at T , is either absorbed (and thus reemitted since T remains constant), or reflected. For instance for a di using body one writes :

received power = emitted power + reflected power

which implies for the luminances :

L0(ν, T ) = L(ν, T ) + [L0(ν, T ) − L(ν, T )]

(9.60)

L0(ν, T ) = L(ν, T ) + [1 − a(ν, T )]L0(ν, T )

This is another way to obtain (9.59).

These types of balances are comparisons between the body luminance L and the black body luminance L0 : if the interpretation of the L0 expression requires Quantum Mechanics, the comparisons of luminances are only based on balances and were discovered during the 19th century.

More generally speaking, the exchanges of energy are taking place according to three processes : conduction, convection and radiation. Whereas the first two mechanisms require a material support, radiation which occurs through photons can propagate through vacuum : it is thanks to radiation that we are in contact with the universe. Through the intersideral vacuum, stars are illuminating us and we can study them by analyzing the received radiation. Our star, the sun, is sending a power of the order of a kW to each 1 m2 surface of earth. The researches on solar energy try to make the best use of this radiation, using either thermal sensors, or photovoltaic cells (“solar cells”) (see §8.5.2) which directly convert it into electricity.

9.2.7Greenhouse E ect

Before interpreting the radiative balance of the earth, let us consider the greenhouse e ect which allows one to maintain inside a building, made of

glass formerly, often of plastics nowadays, a temperature higher than the one outside and thus favors particular vegetables or flowers cultivation (Fig. 9.5). The origin of the e ect arises from the di erence in absorption properties of the building walls for the visible and infrared radiations.

In the visible range, where the largest part of the radiation emitted by the sun is found, the walls are transparent, so that the sun radiation can reach the ground, where it is absorbed. The ground in turn radiates like a black body, with its frequency maximum in the infrared [Wien law, equation (9.41)].

P0 = σSTG4 (1 − a)

if S is the ground surface and a the absorption coe cient, by the walls, of the ground infrared radiation. In the absence of greenhouse, the ground temperature would be TG, such that

P0 = σSTG4

The temperature TG inside the greenhouse is higher than TG since

Here we assumed a perfect transparency of the walls in the visible and no reflection by the walls both in the visible and the infrared.

“The greenhouse e ect” related to a possible climate global warming is an example of radiative balance applied to the earth. The earth is surrounded by a layer of gases, some of which corresponding to molecules with a permanent dipolar moment, due to their unsymetrical arrangement of atoms : this is the case for H2O, CO2, O3, etc. Thus the vibration and rotation spectra of these molecules evidence an infrared absorption.

The sun illuminates the earth with thermal radiation, the maximum of its spectrum being located in the visible [Eq. (9.41)]; this radiation is warming

the ground. The earth, of temperature around 300 K, reemits thermal radiation toward space, the frequency maximum of this reemission is in the infrared. A fraction of the earth radiation is absorbed by the gases around the earth, which in turn reemit a fraction of it, and so on. Finally, according to the same phenomenon as the one described above for a greenhouse, the earth receives a power per unit surface larger than the one it would have reached it if there had only been vacuum around it; its average temperature is 288 K (instead of 255 K if it had been surrounded by vacuum).

The global climate change, possibly induced by human activity, is related to an increase in concentration of some greenhouse e ects gases (absorbing fluorocarboned compounds, which are now forbidden for use, and mainly CO2). If the absorption coe cient increases, the ground temperature will vary in the same way. This will produce e ects on the oceans level, etc. . . 1

More advanced studies on the greenhouse e ect account for the spectra of absorption or transparency of the gases layer around the earth according to their chemical species, etc.

In our daily life, we are surrounded by radiating objects like heating radiators, incandescent lamps. On the contrary we wish other objects to maintain their energy without radiating : this is the case, in particular, for thermos bottles, Dewar cans. We foresee here a very vast domain with technological, ecological and political involvements!

1The Intergovernmental Panel on Climate Change, established by the World Meteorological Organization and the United Nations Environment program, is assessing the problem of potential global climate change. An introduction brochure can be downloaded from its website (www.ipcc.ch).

Summary of Chapter 9

The chemical potential µ of material non-interacting particles following the Bose-Einstein statistics is located below their fundamental state ε1.

The occupation factor by material bosons of an energy level ε is equal to

f (ε) =

1

eβ(ε−µ) − 1

At given density N/Ω, below a temperature TB , which is a function of N/Ω, the Bose condensation takes place : a macroscopic number of particles then occupies the fundamental state ε1, that is, the same quantum state. The thermodynamical properties in this low temperature range are mainly due to the particles still in the excited states.

The superfluidity of helium 4, the superconducting state of some solids, are examples of systems in the condensed Bose state, which is observed for T < TB .

Photons are bosons in non-conserved number, their Bose-Einstein distribution

is given by

1

eβε − 1

The Planck law expresses the spectral density in energy or frequency of photons in an empty cavity of volume Ω, in the frequencies range [ν, ν + dν] :

8πh ν3dν dU = Ωu(ν)dν = Ω c3 eβhν − 1

This law gives the spectral distribution of the “black body” radiation, i.e., of thermal radiation at temperature T = 1/kB β.

The black body is the perfect thermal emitter, its luminance is proportional to u(ν). The thermal emission of a body heated at temperature T corresponds to a luminance at maximum equal to that of the black body.

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General Method for Solving Exercises and Problems

At the risk of stating obvious facts, we recall here that the text of a statistical physics exercise or problem is very precise, contains all the required information and that each word really matters.

Therefore it is recommended to first read the text with the highest attention.

The logic of the present course will always be apparent, i.e, you have to ask yourself the two following questions, in the order given here :

1.What are the microstates of the studied system?

2.Which among these microstates are achieved in the problem conditions?

1.The first question is generally solved through Quantum Mechanics, which reduces to Classical Mechanics in special cases (the ideal gas, for example).

Mostly this is equivalent to the question : “What are the energy states of the system?” In fact they are often specified in the text.

At this stage there are three cases :

–distinguishable particles (for example of fixed coordinates) : see the first part of the course

–classical mobile particles : see the chapter on the ideal gas

–indistinguishable particles : see the second part of the course, in which the study is limited to the case of independent particles.

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Units and physical constants

(From R. Balian, “From microphysics to macrophysics : methods and applications of statistical physics”, Springer Verlag Berlin (1991))

We use the international system (SI), legal in most countries and adopted by most international institutions. Its fundamental units are the meter (m), the kilogram (kg), the second (sec), the ampere (A), the kelvin (K), the mole (mol), and the candela (cd), to which we add the radian (rad) and the steradian (sr).

The derived units called by specific names are : the hertz (Hz = s−1), the newton (N = m kg sec−2), the pascal (Pa = N m−2), the joule (J = N m), the watt (W = J sec−1), the coulomb (C = A s), the volt (V = W A−1), the farad (F = C V−1), the ohm (Ω = V A−1), the siemens (S = A V−1), the weber (Wb = V s), the tesla (T = Wb m−2), the henry (H = Wb A−1), the Celsius degree (C), the lumen (lm = cd sr), the lux (lx = lm m−2), the becquerel (Bq = sec−1), the gray (Gy = J kg−1) and the sievert (Sv = J kg−1).

The multiples and submultiples are indicated by the prefixes deca (da = 10), hecto (h = 102), kilo (k = 103); mega (M = 106), giga (G = 109), tera (T = 1012), peta (P = 1015), exa (E = 1018); deci (d = 10−1), centi (c = 10−2), milli (m = 10−3); micro (µ = 10−6), nano (n = 10−9), pico (p = 10−12), femto (f = 10−15), atto (a = 10−18).

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