STATISTICAL THEORY OF HEAT

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2 STATISTICAL THEORY OF HEAT TIAN MA AND SHOUHONG WANG Abstract. I this paper, we preset a ew statistical theory of heat based o the statistical theory of thermodyamics ad o the recet developmets of quatum physics. Oe motivatio of the theory is the lack of physical carriers of heat i classical thermodyamics. Aother motivatio is the recet discovered photo cloud structure of electros. This leads to a atural cojugate relatio betwee electros ad photos, remiiscet of the cojugate relatio betwee temperature ad etropy. The ew theory i this paper provides a atural coectio betwee these two cojugate relatios: at the equilibrium of absorptio ad radiatio, the average eergy level of the system maitais uchaged, ad represets the temperature of the system; at the same time, the umber desity) of photos i the sea of photos represets the etropy etropy desity) of the system. The theory cotais four parts: 1) the photo umber formula of etropy, 2) the eergy level formula of temperature, 3) the temperature theorem, ad 3) the thermal eergy formula. I particular, the photo umber etropy formula is equivalet to the Boltzma etropy formula, ad, however, possesses ew physical meaig that the physical carrier of heat is the photos. Cotets 1. Itroductio 2 2. Priciples of Statistical Physics Potetial-descedig priciple Statistical distributios 9 3. Quatum Physics Foudatios 13 Date: August 26, Mathematics Subject Classificatio. 86A05, 82B05, 82B10. Key words ad phrases. eergy level formula of temperature, photo umber formula of etropy, thermal eergy formula, etropy, potetial-descedig priciple, Maxwell-Boltzma distributio, Fermi-Dirac distributio, Bose-Eistei distributio. The work was supported i part by the US Natioal Sciece Foudatio NSF), the Office of Naval Research ONR) ad by the Chiese Natioal Sciece Foudatio. 1

3 2 MA AND WANG 3.1. Weakto model of elemetary particle Photo cloud model of electros Photo absorptio ad radiatio mechaism of electros Eergy levels of particles Eergy Level Formula of Temperature Derivatio of temperature formula Physical meaig of the temperature formula Theory of Etropy Physical meaig of etropy Photo umber formula of etropy Law of Temperature Nature of Heat Thermal eergy Balace betwee temperature ad etropy Zeroth law of thermodyamics Caloric theory of heat 35 Refereces Itroductio The mai objective of this paper is to establish a ew theory of heat, based o the statistical theory of thermodyamics ad o the recet developmets of quatum physics. The theory cotais four closely related parts: 1) photo umber formula of etropy, 2) eergy level formula of temperature, 3) a temperature theorem, ad 4) thermal eergy formula. Hereafter first we address the mai motivatios ad mai results of the theory, ad we recapitulate the eeded theoretical foudatios o statistical physics ad quatum physics. Motivatios. The mai motivatios of the ew theory are three-fold. First, as we kow the curret accepted theory of heat is also called the mechaical theory of heat, which related the heat with mechaical work. The theory was first itroduced i 1798 by Bejami Thompso, ad further developed by such great scietists as Sadi Carot, Rudolf Clausius, ad James Clerk Maxwell. This is further developed ito the moder theory of thermodyamics ad statistical physics; see amog may others [18, 22, 2, 19, 8].

4 STATISTICAL THEORY OF HEAT 3 However oe of the remaiig puzzlig questio is how the heat is trasferred. As we kow heat trasfer is classified ito various mechaisms, such as thermal coductio, thermal covectio, thermal radiatio, ad trasfer of eergy by phase chages. However, the ature of heat trasfer is still ot fully uderstood. To be precise, by classical thermodyamics, for a thermodyamic system, the thermal eergy Q 0 is give by 1.1) Q 0 = ST. Here T is the temperature of the system ad S is the etropy give by the Boltzma formula: 1.2) S = k l W, where k is the Boltzma costat, ad W is the umber of microscopic cofiguratios of the system. It is the clear that i moder thermodyamics, there is simply o physical heat 1.3) carrier i both the temperature T ad the etropy S, ad hece there is o physical carrier for thermal eergy Q 0 = ST. Historically, the caloric theory of heat was developed for such a purpose, ad becomes ow a obsolete theory. Basically, the caloric theory says that heat is made up of a fluid called caloric that is massless ad flows from hotter bodies to colder bodies. It was cosidered that caloric was a massless gas that exists i all matter, ad is coserved. However the trasfer betwee heat ad mechaical work makes the caloric theory obsolete. Kow physical facts show that the essece of thermal radiatio is the absorptio ad radiatio of photos, ad as we shall see below, the ew statistical theory of heat makes precise that photos are ideed the eeded physical carrier of heat. The secod motivatio of the study is the recet developmets i particle physics, which reveal the photo cloud structure of electros; see Sectio 3.2, as well as [13] for details. It is show that there is a attractig shell regio of weak iteractio betwee the aked electro ad photos as i Figure 1.1. Sice photos carry weak charges, they are attached to the electro i the attractig shell regio formig a cloud of photos. The attractig shell regio i Figure 1.1 of a electro results i the ability for the electro to attract ad emit photos. A macroscopic system is immersed i a sea of photos mediators). Whe a photo eters the attractig shell regio of a electro, it will be absorbed by the electro. A electro emits photos as its velocity chages, which

5 4 MA AND WANG Figure 1.1. A cloud of photos forms i the regio ρ 1 <r<ρ 2 ear the aked electro. is called the bremsstrahlug. Also, whe the orbitig electro jumps from higher eergy level to a lower eergy level, it radiates photos. Hece electros i the system are costatly i a state of absorbig ad emittig photos, resultig chages o their eergy levels. I other words, we reach the followig physical coclusio: i the microscopic world, electros ad photos form a atural cojugate pair of physical carriers for emissio ad ab- 1.4) sorptio. This physical coclusio leads to the third motivatio of the theory of heat developed i this paper. Namely, i view of 1.1), heat is attributed to the cojugate relatio betwee temperature ad etropy. Hece by 1.4), a theory of heat has to make coectios betwee the two cojugate relatios: cojugatio betwee electros ad photos 1.5) cojugatio betwee temperature ad etropy. The ew theory i this paper provides precisely such a coectio: at the equilibrium of absorptio ad radiatio, the average eergy level of the system maitais uchaged, ad represets the temperature of the system; at the same time, the umber desity) of photos i the sea of photos represets the etropy etropy desity) of the system. Mai results We ow ready to address briefly the four parts of the statistical theory of heat developed i this paper. 1. Eergy level formula of temperature. I view of the relatio 1.5), temperature must be associated with the eergy levels of electros,

6 STATISTICAL THEORY OF HEAT 5 sice it is a itesive physical quatity measurig certai stregth of heat, remiiscet of the basic characteristic of eergy levels of electros. Also otice that there are abudat orbitig electros ad eergy levels of electros i atoms ad molecules. Hece the eergy levels of orbitig electros, together with the kietic eergy of the system particles, provide a truthful represetatio of the system particles. We derive the followig eergy level formula of temperature usig the well-kow Maxwell-Boltzma, the Fermi-Dirac, ad the Bose- Eistei distributios: 1 a ) a ε for classical systems, N N1 + β l ε ) 1.6) kt = 1+ a ) a ε for Bose systems, g 1 a g N1 + β l ε ) ) a ε N1 + β l ε ) for Fermi systems. Here ε are the eergy levels of the system particles, N is the total umber of particles, g are the degeeracy factors allowed quatum states) of the eergy level ε,ada are the distributios, represetig the umber of particles o the eergy level ε. If we view ε = 1 a ε N as the average eergy level for the thermodyamical system, the the above formulas amout to sayig that temperature is simply the weighted) average eergy level of the system. Formulas 1.6) eable us to have a better uderstadig o the ature of temperature. I summary, the ature of temperature T is the weighted) average eergy level. Also, the temperature T is a fuctio of distributios {a } ad the eergy levels {ε } with the parameters {β } reflectig the property of the material. From the temperature formula, we ca easily see that for a thermodyamic system with temperature at absolute zero, all particles fill the lowest eergy levels. Also, it is ot hard to see from the temperature formula the existece of highest temperature. 2. Photo umber formula of etropy. I view of 1.5), sice etropy S is a extesive variable, we eed to characterize etropy as the umber of photos i the photo gas betwee system particles, or

7 6 MA AND WANG the photo desity of the photo gas i the system. Also, photos are Bosos ad obey the Bose-Eistei distributio. The we ca make a coectio betwee etropy ad the umber of photos ad derive 1.7) S = kn 0 [ 1+ 1 kt ε a N 0 where ε are the eergy levels of photos 1,ada are the distributio of photos at eergy level ε, N 0 = a is the total umber of photos betwee particles i the system, ad ε a kt represets the umber of photos i the sese of average eergy level. It is worth metioig that this ew etropy formula is equivalet to the Boltzma etropy formula 1.2). However, their physical meaigs have chaged: the ew formula 1.7) provides explicitly that 1.8) the physical carrier of heat is the photos. 3. Temperature theorem. By the temperature ad the etropy formulas 1.6) ad 1.7), we arrive immediately at the followig results of temperature, stated i Theorem 5.2 as the law of temperature: 1) There are miimum ad maximum values of temperature with T mi =0ad T max beig give by 4.33); 2) Whe the umber of photos i the system is zero, the temperature is at absolute zero; amely, the absece of photos i the system is the physical reaso causig absolute zero temperature; 3) Nerst Theorem) With temperature at absolute zero, the etropy of the system is zero; 4) With temperature at absolute zero, all particles fills all lowest eergy levels. 4. Thermal eergy formula. Thaks to the etropy formula 1.7), we derive immediately the followig thermal eergy formula: 1.9) Q 0 = ST = E 0 + kn 0 T, where E 0 = a ε is the total eergy of photos i the system, ε are the eergy levels of photos, ad a are the distributio of photos at eergy level ε,adn 0 is the umber of photos i the system. Statistical physics ad quatum physics foudatios 1 We emphasize here that for brevity we use the same ε to deote, respectively, the eergy levels for photos i 1.7), ad the eergy levels for system particles electros) i 1.6). ],

8 STATISTICAL THEORY OF HEAT 7 The theory of heat preseted i this paper is established based o physical theories o fudametal iteractios, the photo cloud model of electros, the first law of thermodyamics, statistical theory of thermodyamics, radiatio mechaism of photos, ad eergy level theory of micro-particles. The theory utilizes rigorous mathematics to reveal the physical essece of temperature, etropy ad heat. 1. Statistical physics foudatio. I derivig the temperature ad the etropy formulas 1.6) ad 1.7), we make a direct use of the Maxwell-Boltzma MB), the Fermi-Dirac FD), ad the Bose-Eistei BE) distributios. These distributios are respectively for classical systems, the Fermi systems, ad the Bose systems. I a recet paper [17], the authors postulated the potetial-descedig priciple PDP). We show that PDP is a more fudametal priciple tha the first ad secod laws i thermodyamics, gives rise to dyamical equatios for o-equilibrium systems, ad serves as the first priciple to describe irreversibility of all thermodyamic systems. Also, together with the Boltzma etropy formula 1.2) ad the classical priciple of equal probability stated i Priciple 2.2, PDP leads to all three distributios. Hece for the ew theory of heat, the eeded statistical physics foudatio is the Boltzma etropy formula, priciple of equal probability ad the potetial descedig priciple. 2. Quatum physics foudatio. It is clear that a theory of heat depeds o the quatum behavior of basic microscopic costituets of matter. We refer iterested readers to [24, 5, 4, 6, 13] for moder theory of quatum mechaics ad particle physics. The followig recet developmets i quatum physics play a crucial role for our study i this paper: 1) weak iteractio force formula, 2) weakto model of elemetary particles, 3) photo cloud structure of electros, 4) photo absorptio ad radiatio mechaism of electros, ad 5) eergy levels of micro-particles. These quatum physics foudatios are recapitulated i Sectio 3, ad we also refer the iterested readers to [13] ad the refereces therei for more details. The paper is orgaized as follows. Sectio 2 recalls the potetialdescedig priciple PDP) ad the three basic statistics: the Maxwell- Boltzma distributio, the Fermi-Dirac distributio ad the Bose- Eistei distributio. Sectio 3 recapitulates quatum physics basis eeded for the theory of heat. I Sectios 4 ad 5 we establish the

9 8 MA AND WANG mai compoets of the theory, ad Sectio 6 addresses the ature of heat based o the ew theory. 2. Priciples of Statistical Physics I this sectio, we itroduce some basic priciples for statistical physics, servig as the statistical foudatio of the statistical theory of heat to be developed i Sectios 4-6. We refer iterested readers to [20, 21, 9, 12, 11, 7, 3, 23, 1] amog may others for classical theories of statistical physics. The potetial-descedig priciple is itroduced by the authors i [17] Potetial-descedig priciple. For a give thermodyamic system, the order parameters state fuctios) u =u 1,,u N ), the cotrol parameters λ, ad the thermodyamic potetial or potetial i short) F are well-defied quatities, fully describig the system. The potetial is a fuctioal of the order parameters, ad is used to represet the thermodyamic state of the system. There are four commoly used thermodyamic potetials: the iteral eergy, the Helmholtz free eergy, the Gibbs free eergy, ad the ethalpy. After a thorough examiatio of thermodyamics, we discovered i [17] that the followig Potetial-Descedig Priciple PDP) is a fudametal priciple i statistical physics. Priciple 2.1 Potetial-Descedig Priciple). For each thermodyamic system, there are order parameters u =u 1,,u N ), cotrol parameters λ, ad the thermodyamic potetial fuctioal F u; λ). For a o-equilibrium state ut; u 0 ) of the system with iitial state u0,u 0 )= u 0, we have the followig properties: 1) the potetial F ut; u 0 ); λ) is decreasig: d dt F ut; u 0); λ) < 0 t >0; 2) the order parameters ut; u 0 ) have a limit lim ut; u 0)=ū; t 3) there is a ope ad dese set O of iitial data i the space of state fuctios, such that for ay u 0 O, the correspodig ū is a miimum of F, which is called a equilibrium of the thermodyamic system: δfū; λ) =0.

10 STATISTICAL THEORY OF HEAT 9 We have show that PDP is a more fudametal priciple tha the first ad secod laws, ad provides the first priciple for describig irreversibility, leads all three distributios: the Maxwell-Boltzma distributio, the Fermi-Dirac distributio ad the Bose-Eistei distributio i statistical physics. Cosequetly, the potetial-descedig priciple is the first priciple of statistical physics. Also importatly, based o PDP, the dyamic equatio of a thermodyamic system i a o-equilibrium state takes the form 2.1) du = AδF u, λ) for isolated systems, dt du = AδF u, λ)+bu, λ), dt 2.2) for coupled systems, AδF u, λ) Bu, λ) =0 where δ is the derivative operator, B represets couplig operators, ad A is a symmetric ad positive defiite matrix of coefficiets. We refer iterested readers to [17] for details Statistical distributios. For a thermodyamic system, a mai compoet of statistical theory is to study the probability distributio of particles i differet eergy levels of the system i equilibrium. I this sectio, we aim to derive all three distributios: the Maxwell- Boltzma distributio, the Fermi-Dirac distributio ad the Bose- Eistei distributio, based oly o 1) the priciple of equal a priori probabilities, ad 2) the potetial-descedig priciple PDP). The latter was stated i the last sectio, ad the priciple of equal a priori probabilities ca be stated as follows: Priciple 2.2 Priciple of Equal Probability PEP)). A equilibrium thermodyamic system has a equal probability of beig i ay microstate that is cosistet with its curret macrostate Maxwell-Boltzma statistics. Now cosider a isolated classical thermodyamic system, the total eergy E ad the total umber of particles N are costats. Each particle i the system must be situated o a eergy level ε, ad the total eergy is fiite. Sice the system is i equilibrium, we have 2.3) a def = [umber of particles o ε ] = [probability at ε ] N. The PEP esures the time-idepedece of a. Wehavethethe followig arragemet of umber of particles o differet eergy levels:

11 10 MA AND WANG ε 1 < ε 2 < < ε NE, 2.4) g 1 g 2 g NE, a 1 a 2 a NE, where g represets the degeeracy factor allowed quatum states) of the eergy level ε. For a isolated system, 2.5) N = a = costat, E = a ε = costat. It is clear that the multiplicity fuctio W for the distributio 2.4) is give by: 2.6) W = W a 1,,a NE ). The aim is the to fid the relatios betwee ε ad a uder costrait 2.5): 2.7) a = fε,t) for 1 N E. For a isolated thermodyamic system, the temperature T is a cotrol parameter, ad cosequetly, its thermodyamic potetial fuctioal is 2.8) F = E ST, where E is the total eergy, ad S is the etropy. Classically, the etropy is give by the famous Boltzma formula: 2.9) S = k l W, where k = J/K is the Boltzma costat. By the potetial-descedig priciple, Priciple 2.1, the distributio {a } at the thermodyamic equilibrium solves the followig miimal potetial variatioal equatios of the potetial fuctioal 2.8): δ [ ] 2.10) kt l W + α0 a + β 0 a ε =0, δa where α 0 ad β 0 are the Lagragia multipliers of costraits 2.5). I view of the distributio of particles 2.4), the multiplicity fuctio W is give by 2.11) W = N! g a, a! By the Stirlig formula k! =k k e k 2πk,

12 ad usig l2πk) k, wehave STATISTICAL THEORY OF HEAT 11 l W = N l N N a l a + a + a l g. Sice a = N, wearriveat l W = N l N a l a g, which implies from 2.10) that l a + α + βε =0 for1 N E. g This gives rise to the famous Maxwell-Boltzma distributio: 2.12) a = N Z g e ε/kt. Here Z is the partitio fuctio defied by 2.13) Z = g e βε, Also, direct computatio shows that the MB distributio is ideed the miimal poit of the potetial fuctioal 2.8) uder the costraits 2.5). The partitio fuctio Z defied by 2.13) is aother importat thermodyamical quatity i statistical mechaics. I fact, oce we kow the detailed expressio of the partitio fuctio, we ca derive other related thermodyamical quatities as follows: 2.14) U = N β l Z iteral eergy, S = Nk l Z β ) β l Z etropy, f = N β X l Z geeralized force, F = NkT l Z potetial fuctioal Bose-Eistei distributio. The Maxwell-Boltzma statistics is for classical systems of particles. For systems where quatum behavior is promiet, quatum statistics is the ultimately eeded. Quatum statistics cosists of the Bose-Eistei BE) statistics for systems of bosoic particles, ad the Fermi-Dirac statistics for systems of fermioic particles.

13 12 MA AND WANG The goal here is the same as the statistics for classical particle systems: to fid the relatios 2.7) betwee ε ad a uder costrait 2.5). For a quatum system of bosoic particles, the multiplicity fuctio associated with 2.4) is 2.15) W BE = g + a 1)! a!g 1)!. As for the Maxwell-Boltzma distributio, solvig 2.10) leads to the followig Bose-Eistei distributio: 2.16) a = g e ε μ)/kt 1, where μ is the chemical potetial. For a quatum system of bosoic particles, the partitio fuctio is give by 2.17) Z = 1 e α βε ) g, α = μ kt, β = 1 kt. Also we ca derive other related thermodyamical quatities, the total umber of particles N, the total eergy E, the geeralized force f, the pressure p, the etropy S, the free eergy F, ad the Gibbs eergy G, i terms of the partitio fuctio Z as follows: 2.18) N = l Z, α E = l Z, β f = 1 l Z, β X p = 1 l Z, β V S = k [ l Z α F = E ST = kt α + β β G = NkTα = ktα l Z. α ) ] l Z, l Z α α l Z ),

14 STATISTICAL THEORY OF HEAT Fermi-Dirac distributio. Fermios obey the Pauli exclusio priciple. Hece for a quatum system of fermios, the multiplicity fuctio is give by 2.19) W FD = g! a!g a )!. The it is easy to fid the followig Fermi-Dirac distributio 2.7) betwee ε ad a uder costrait 2.5): g 2.20) a = e ε μ)/kt +1, where μ is the chemical potetial. Also the partitio fuctio for a Fermi system is give by 2.21) Z = 1+e α βε ) g, α = μ kt, β = 1 kt. The relatios betwee the partitio fuctio ad other thermodyamical quatities are give by 2.18) as well. 3. Quatum Physics Foudatios We itroduce i this sectio some recet developmets i quatum physics, which serve as the quatum physics foudatios for the statistical theory of heat that we itroduce i this paper. For a more detailed accout of these recet developmets, see [13] Weakto model of elemetary particle. The weakto model of elemetary particle was first itroduced by the authors [14, 13]. This theory proposes six elemetary particles, which we call weaktos, ad their ati-particles: 3.1) w, w 1, w 2, ν e, ν μ, ν τ, w, w 1, w 2, ν e, ν μ, ν τ, where ν e,ν μ,ν τ are the three geeratio eutrios, ad w,w 1,w 2 are three ew particles, which we call w-weaktos. These are all massless particles with spi J = 1. Each of them carries a weak charge, ad 2 oly w ad w carry a strog charge. Also, the eutrios do ot carry electric charge, w carries 2/3 electric charge, w 1 carries 1/3 electric charge ad w 2 carries 2/3 electric charge. 1) Weakto costituets of particles

15 14 MA AND WANG The weakto costituets of charged leptos ad quarks are give by e = ν e w 1,w 2, μ = ν μ w 1 w 2, τ = ν τ w 1 w 2, 3.2) u = w w 1 w 1, c = w w 2 w 2, t = w w 2 w 2, d = w w 1 w 2, s = w w 1 w 2, b = w w 1 w 2, where c, t ad d, s, b are distiguished by the spi arragemets. The weakto costituets of the mediators ad their dual mediators are give by γ =cosθ w w 1 w 1 si θ w w 2 w 2, ) vector photo, γ 0 =cosθ w w 1 w 1 si θ w w 2 w 2, ) scalar photo, 3.3) g k = w w, ), vector gluos, g0 k = w w, ) scalar gluos The ν-mediator ν has spi-0 with the followig weakto costituets: 3 3.4) ν = α 1 ν e ν e + α 2 ν μ ν μ + α 3 ν τ ν τ ), αl 2 =1. Each gluo carries two strog charges ad two weak charges, ad participates both the weak ad strog iteractios. Both photo ad the ν mediator oly carry respectively two weak charges, ad participate the weak iteractio, but ot the strog iteractio. All three mediators carry o electric charge. 2) Mass geeratio mechaism For a particle movig with velocity v, itsmassm ad eergy E obey the Eistei relatio 3.5) E = mc2. 1 v2 c 2 Usually, we regard m asastaticmasswhichisfixed,adeergyisa fuctio of velocity v. Now, takig a opposite viewpoit, we regard eergy E as fixed, mass m as a fuctio of velocity v, ad the relatio 3.5) is rewritte as 3.6) m = 1 v2 E c 2 c. 2 Thus, 3.6) meas that a particle with a itrisic eergy E has zero mass m = 0 if it moves at the speed of light v = c, ad will possess ozero mass if it moves with a velocity v<c. l=1

16 STATISTICAL THEORY OF HEAT 15 All particles icludig photos ca oly travel at the speed sufficietly close to the speed of light. Based o this viewpoit, we ca thik that if a particle movig at the speed of light approximately) is decelerated by a iteractio force F,obeyig dp dt = 1 v2 F, c 2 the this massless particle will geerate mass at the istat. I particular, by this mass geeratio mechaism, several massless particles ca yield a massive particle if they are boud i a small ball, ad rotate at velocities less tha the speed of light. For the mass problem, we kow that the mediators: 3.7) γ, g k,ν ad their dual particles, have o masses. To explai this, we ote that these particles i 3.3) ad 3.4) cosist of pairs as 3.8) w 1 w 1, w 2 w 2, w w, ν l ν l. The weakto pairs i 3.8) are boud i a circle with radius R 0 as show i Figure 3.1. Sice the iteractig force o each weakto pair is i the directio of their coectig lie, they rotate aroud the ceter 0 without resistace. As F = 0 i the movig directio, by the relativistic motio law: d 3.9) P dt = 1 v2 F, c 2 the massless weaktos rotate at the speed of light. Hece, the composite particles formed by the weakto pairs i 3.8) have o rest mass. Figure 3.1 For the massive particles 3.10) e, μ, τ, u, d, s, c, t, b,

17 16 MA AND WANG by 3.2), they are made up of weakto triplets with differet electric charges. Hece the weakto triplets are arraged i a irregular triagle as show i Figure 3.2. Cosequetly, the weakto triplets rotate with ozero iteractig forces F 0 from the weak ad electromagetic iteractios. By 3.9), the weaktos i the triplets at a speed less tha the speed of light due to the resistace force. Thus, by the mass geeratig mechaism above, the weaktos become massive. Hece, the particles i 3.10) are massive. Figure Photo cloud model of electros Photo clouds of electros. The weakto costituets of a electro are ν e w 1 w 2. Therefore, a electro carries three weak charges, which are the source of the weak iteractio. A photo has the weakto costituets as give by 3.3), ad carries two weak charges. The weak force formula betwee the aked electro ad a photo γ is give by; see [13]: F = g w ρ γ )g w ρ e ) d [ 1 dr r e kr B ] 3.11) ρ 1+2kr)e 2kr, =g w ρ m )g w ρ e )e kr [ 1 r rr 0 4B ρ r r 2 0 e kr ], where k =1/r 0 =10 16 cm 1,g w ρ m )adg w ρ e ) are the weak charges of mediators ad the aked electro, expressed as ) 3 ) 3 ρw ρw g w ρ γ )=2 g w, g w ρ e )=3 g w, ρ γ ad B/ρ is a parameter determied by the aked electro ad the photo. ρ e

18 STATISTICAL THEORY OF HEAT 17 By the weak force formula 3.11), there is a attractig shell regio of weak iteractio betwee aked electro ad the photo 3.12) F<0 for ρ 1 <r<ρ 2, as show i Figure 1.1, which is reproduced i Figure 3.3 for coveiece. Sice photos carry weak charges, they are attached to the Figure 3.3. Electro structure. electro i the attractig shell regio 3.12), formig a cloud of photos. The irregular triagle distributio of the weaktos ν e,w 1,w 2 geerate a small momet of force o the mediators. Meawhile there also exist weak forces betwee them. Therefore the bosos will rotate at a speed less tha the speed of light, ad geerate a small mass attached to the aked electro ν e w 1 w Agular mometum rule. The Agular Mometum Rule 3.1 below was first discovered i [14, 13, 15]. It esures that the photos i the clouds of electros ca oly be scalar photos J = 0, ad cosequetly the photo cloud of a electro does ot chage the spi of the electro J =1/2. Agular Mometum Rule 3.1. Oly the fermios with spi J = 1 2 ad the bosos with J =0ca rotate aroud a ceter with zero momet of force. The particles with J 0, 1 will move o a straight lie uless 2 there is a ozero momet of force preset. Also we remark that based o the mechaism of decay ad scatterig of particles, weakto exchages may occur durig the followig γ-γ scatterig process, leadig to the trasformatio betwee scalar photos ad vector photos: γ + γ γ + γ,

19 18 MA AND WANG ad the correspodig weakto costituet exchage is give by w 1 w 1 )+w 1 w 1 ) w 1 w 1 )+w 1 w ). This observatio shows that although the photos i the photo cloud of a electro ca oly scalar photos, both scalar ad vector photos are abudat i Nature Photo absorptio ad radiatio mechaism of electros. The attractig shell regio i Figure 3.3 of a electro results i the ability for the electro to attract ad emit photos. A macroscopic system is immersed i a sea of photos mediators). Whe a photo eters the attractig shell regio of a electro, it will be absorbed by the electro. A electro emits photos as its velocity chages, which is called the bremsstrahlug. Also, whe the orbitig electro jumps from higher eergy level to a lower eergy level, it radiates photos. Hece electros i the system are costatly i a state of absorbig ad emittig photos, resultig chages o their eergy levels. As we shall see i the ext sectio, at the equilibrium of absorptio ad radiatio, the average eergy level of the system maitais uchaged, ad represets the temperature of the system; at the same time, the umber desity) of photos i the sea of photos represets the etropy etropy desity) of the system. The reasos why bremsstrahlug ca occur is ukow i classical theories. Based o the electro structure theory i Sectio 3.2, this pheomeo ca be easily explaied. Figure 3.4. a) The aked electro is accelerated or decelerated i a electromagetic field; ad b) the mediators photos) fly away from the attractig shell regio uder a perturbatio of momet of force. I fact, if a electro is situated i a electromagetic field, the the electromagetic field exerts a Coulomb force o the aked electro ν e w 1 w 2, but ot o the attached eutral mediators. Thus, the aked electro chages its velocity, which draws the mediator cloud to move

20 STATISTICAL THEORY OF HEAT 19 as well, causig a perturbatio to momet of force o the mediators. As the attractig weak force i the shell regio 3.6) is small, uder the perturbatio, the cetrifugal force makes some mediators i the cloud, such as photos, flyig away from the attractig shell regio, ad further accelerated by the weak repellig force outside this shell regio to the speed of light, as show i Figure Eergy levels of particles. This sectio is based o [16, 13]. The mass m, eergye ad the mometum p of a particle obey the Eistei eergy-mometum relatio; see amog may others [10, 13]: 3.13) E 2 = m 2 c 4 + c 2 p 2. There are differet eergy levels, which ca udergo chages by 1) absorbig ad/or emittig photos, ad 2) exchagig the iterior costituets. The weaktos are elemetary particles, ad all other particles are composite. For composite particles, the eergy levels determied by their costituets are called itrisic eergy levels, which ca chage through exchagig costituets. Eergy levels of particles play a importat role i statistical physics. There are may particles i Nature, ad we are iterested i eergy levels of the followig particles, which play crucial role i statistical physics: 3.14) photos, electros, ad atoms. Hereafter we focus o the eergy levels of these particles Eergy levels of photos. The weakto costituets of a photo are the followig two weaktos, symmetrically bouded together by the weak force: γ = w i w i i =1, 2. It suffices for us to cosider the bouded states of oe weakto. As the weakto w i i =1, 2) are massless, the wave fuctio describig them is the two-compoet Weyl spior: 3.15) ψ =ψ 1,ψ 2 ), ad the correspodig wave equatios are ψ 3.16) σ D) t = c σ D) 2 ψ ig w 2 { σ D),A 0 }ψ, where {A, B} = AB + BA is the ati-commutator, σ =σ 1,σ 2,σ 3 )is the Pauli matrix operator, the operator D is defied by 3.17) D = + i g w c W,

21 20 MA AND WANG with W μ =W 0, W ) berig the weak iteractio potetial of weaktos. The spectral equatio of photos ca be derived by settig ) ψ = e iλt/ ϕ1 ϕ, ϕ =, ϕ 2 where λ is the boudig eergy. We ifer from 3.16) that ) 3.18) c σ D) 2 ϕ1 + ig ) ) w ϕ 2 2 { σ D),W ϕ1 0 } = iλ σ ϕ D) ϕ1. 2 ϕ 2 Sice the weaktos are cofied i the photo, we ca set ψ =0 outside of the photo. Cosequetly, we have the followig boudary coditio: 3.19) ϕ =0 for x = ρ γ, where ρ γ is the radius of photos. We ca the derive the followig coclusios for eergy levels of photos usig the above liear eigevalue problem 3.18) ad 3.19). 1) There are fiite umber of egative eigevalues for 3.18) ad 3.19), represetig the boudig eergies of the weaktos: 3.20) <λ 1 λ 2 λ N < 0; 2) There are fiite umber of eergy levels for photos, give by 3.21) E k = E 0 + λ k for 1 k N, where E 0 is the itrisic eergy of the two weakto costituets of the photo. Hece the eergy levels of a photo are fiite: 3.22) 0 <E 1 E 2 E N ; 3) The frequecies of a photo are discrete: 3.23) ω k = E k /, Δω k = ω k+1 ω k =λ k+1 λ k )/ ; 4) The umber of eergy levels of photos ca be estimated as follows: ) Bw ρ γ g 2 3 w 3.24) N = 10 90, β 1 ρ w c ad the eergy differeces ca be estimated as 3.25) ΔE E max E mi N which is small ad uobservable. = λ N λ 1 N ev,

22 STATISTICAL THEORY OF HEAT Eergy levels of electros. The electros are massive with three weakto costituets: ν e w 1 w 2. As metioed earlier, these three weaktos possess differet electric charges, ad are arraged i a irregular triagle as show i Figure 3.2, becomig the massive. Hece they are govered by three Dirac spiors: ψ j =ψ1, j,ψ4), j j =1, 2, 3. It is the easy to derive the eergy level equatio for a electro as follows: 2 + i 2g w 3.26) 2m j c W)2 ϕ j +22g w W 0 + μ j curlw)ϕ j = λϕ j, ϕ j =0 forj =1, 2, 3, x = ρ e, where ρ e is the radius of a electro, W μ =W 0, W) istheweakiteractio potetial, ϕ j =ϕ j 1,ϕ j 2) are the eigestates of the j-th weakto, ad μ j = g w σ 2m j is the weak magetic momet of the j-th weakto. We derive from 3.26) the followig coclusios: 1) The itrisic eergy levels of electros are fiite ad discrete. 2) The umber N of itrisic eergy levels of a electro ca be approximately estimated as N = [ 4 λ 1 B w ρ 2 e ρ w m w c ] gw 2 3/ , c where ρ e is the radius of th electro, ρ w is the radius of the weakto, B w is the weak iteractio parameter i 3.11), m w isthemassof the costituet weaktos of the electro caused by ozero iteractig force from the weak ad the electromagetic iteractios, ad λ 1 is the first eigevalue of Δ. 3) I view of the photo cloud structure that a electro cosists of the aked electro ad the shell-layer of its photos cloud, the total umber N of eergy levels of electros is about N = umber of itrisic eergy levels umber of eergy levels of photo Eergy levels of atoms. Classical eergy levels was developed based o the Bohr atomic models ad the Schrödiger equatios. The is made up of ucleus ad the orbitig electros, the ucleus is made up of protos p ad eutros, which are made up of three quarks:

23 22 MA AND WANG p = uud, = udd. I additio, the weakto costituets of upper addowquarksareu = w w 1 w 1 ad d = w w 1 w 2. Therefore, the eergy levels of a atom is the sum of eergy levels of the ucleus ad the eergy levels of the orbitig electros, ad the eergy levels of the ucleus are determied by eergy levels of a ucleus = Ek 1 + λ 1 j, where Ek 1 are the eergy levels of ucleos, λ1 j are the egative eigevalues of the spectral equatio for the atom, represetig the boudig eergies boudig the ucleos. Therefore, Ek 1 = El 2 + λ 2 j + eergy levels of absorbed mediators of the ucleos. Here El 2 are the eergy levels of quarks, λ 2 j are the egative eigevalues of the spectral equatio of the ucleos, represetig the boudig eergy betwee quarks. Fially, E 2 l = E λ 3 j + eergy levels of the mediators absorbed by quarks, where E0 3 is the itrisic eergy of the weaktos i the quark, ad λ 3 j is the egative eigevalues of the spectral equatio for the quark, represetig the boudig eergies boudig the weaktos iside the quarks Physical coclusios of eergy levels of particles. I summary, we have the followig physical coclusios for eergy levels of particles, which provide the particle physics foudatio of statistical physics. 1) The eergies of micro-particles are o their eergy levels, ad there are fiite umber of eergy levels, which are discrete; 2) Particles ca jump to differet differet eergy levels by a) absorbig or emittig photos, ad b) exchagig their costituet particles. 3) The eergy of a particle obeys the Eistei eergy-mometum relatio 3.13). Whe the eergy level of a particle chages, its mass ad mometum will udergo chages as well. For a fixed eergy level, the mass ad the mometum ca udergo trasformatios betwee each other. 4) The umber N of eergy levels is large, ad the gas betwee adjacet eergy levels are small; they ca be estimated roughly estimated as for photos, umber of eergy levels for electros, for atoms,

24 STATISTICAL THEORY OF HEAT 23 ΔE k = E k+1 E k ev for photos. 4. Eergy Level Formula of Temperature We have itroduced the photo cloud structure of subatomic particles i the previous sectio. I this sectio, we use such structure of sub-atomic particles to reveal the ature of temperature ad etropy. Basically, amog basic costituets of matter, electros, protos ad eutros are fudametally importat. Both protos ad eutros are cofied i the ucleos, ad electros are the oly charged particles abudat iside the matter. The essece of thermal radiatio is the radiatio ad absorptio of photos. With the photo cloud structure of electros, electros ad photos form a pair of cojugate physical carriers for absorptio ad emissio associated with thermal radiatio. O the other had, thermal eergy is the cojugate relatio betwee temperature ad etropy. Hece a correct statistical theory of heat must make a precise coectio of the followig correspodece: 4.1) cojugatio betwee electros ad photos cojugatio betwee temperature ad etropy Derivatio of temperature formula. The mai objective of this sectio is to derive the followig temperature formula: 1 a ) a ε for classical systems, N N1 + β l ε ) 4.2) kt = 1+ a ) a ε for Bose systems, g N1 + β l ε ) 1 a ) a ε for Fermi systems. g N1 + β l ε ) If we view ε = 1 a ε N as the average eergy level for the thermodyamical system, the the above formulas shows that temperature is simply the weighted average eergy level of the system. Hereafter we derive these formulas usig the basic distributios. Classical systems. Cosider a classical equilibrium thermodyamic system with eergy levels of the particles give by 4.3) ε 1, ε 2,, ε NE.

25 24 MA AND WANG By the MB distributio 2.12), the total eergy of the system is 4.4) E = a ε = N g ε e ε/kt, Z where N is the total umber of particles, ad Z = g e ε kt is the partitio fuctio. Whe we fid the total eergy E, we ca view 4.4) as a equatio defiig a implicit fuctio of the temperature T i terms of the eergy levels i 4.3): T = T ε 1,,ε NE ). Physically, it meas that uder the ivariace of the total eergy E, the distributio {a } chages as {ε } vary, leadig to the chage of the temperature T. Hece we ca assume the followig expressio of T : 4.5) T = α T ε ), where the coefficiets α are to be determied. Physically, it is atural to assume that fluctuatios o a specific eergy level ε will oly lead to fluctuatios o the eergy a ε o the level ε i the total eergy 4.6) E = m a mε m. Mathematically, by 4.6), the implicit fuctio relatio ca be determied usig the followig variatio: 0= E [ ε 4.7) = Ng ε ε Z e ε/kt [ 1 = N g Z ε Z Z 2 ε 4.8) Z ε = Cosequetly, we have ] 1 + g ε Z e ε/kt g kt + g ε T kt 2 ε = Z N 0=a a ε Z a kt + a ε kt 2 Z N ) e ε/kt kt + ε T kt 2 ε ) e ε/kt T ε. Z ε a ε kt + a ε 2 kt 2 T ε )],

26 = a + a2 ε N STATISTICAL THEORY OF HEAT 25 1 kt a2 ε 2 T a ε NkT 2 ε kt + a ε 2 T, kt 2 ε which implies the followig differetial equatio for T ε ): T 4.9) = T k 1 a ) 1 T 2. ε ε N ε 2 Let x = ε, the we ifer from 4.9) that 4.10) T = T x k 1 a ) 1 T 2 N x. 2 Now let y def = T x, the T = xy + y. Equatio 4.10) becomes 4.11) xy = k 1 a ) 1 y 2. N Physically, we may assume that a 4.12) N costat. The 4.11) is equivalet to 4.13) 1 a ) dy dx N ky = 2 x. Sice x = ε carries the dimesio of eergy, we write the solutio of 4.13) as follows: 1 a /N ) =l ε + C, ky ε 0 where ε 0 is the uit of eergy. Hece we obtai that ktε) = 1 a ) ε. N C +lε/ε 0 As discussed earlier, all T ε m ) should take the same form, we obtai that kt = 1 a ) α ε. N C +lε /ε 0 Let β =1/C ad θ = α /C,thewehave 4.14) kt = 1 a ) θ ε. N 1+β l ε/ε 0

27 26 MA AND WANG We ow try to determie the coefficiets θ. For this purpose, we first defie the traslatioal derivative with respect to all eergy levels: [ ] T ) = lim T ε 1 +Δε, ε 2 +Δε, ) T ε 1,ε 2, ) Δε 0 Δε = θ 1 a ) 1 β β l ε ). N ε 0 The we take the traslatio derivative o both sides of 4.4). By the physical assumptio 4.6), we obtai that δε = δε for all ad 4.16) 0 = δe = [ ε ] g ε Z e ε/kt δε. Also by 4.6), we have [ ε ] 1 g ε Z e ε/kt =g Z ε Z 4.17) Z 2 ε + g ε Z ) e ε/kt 1 kt + ε kt 2 T ) e ε/kt, where T is as i 4.15), ad by 4.6), we use the followig approximatio for the cotributio of Z/ ε to the -th eergy level: Z 4.18) ε = g 1 kt + ε ) kt T e ε/kt. 2 Hece by 4.16)-4.18), we obtai that 4.19) kt = a 1 a ) ε N N a 1 a ) T N N T ε2. O the other had, 4.20) kt = = [ θ 1 a N {θ [ 1 a N ] ε 1 1+β l ε /ε 0 [ ]ε θ 1 a N ] β ε l ε ε 0 }. We deduce the from 4.19) ad 4.20) that 4.21) θ = a N, β T ε = T lε /ε 0 ), ad cosequetly 4.2) for classical systems follows.

28 STATISTICAL THEORY OF HEAT 27 Quatum systems. For a quatum system, we first recall the Bose- Eistei statistics 2.16) or the Fermi-Dirac statistics 2.20): ) g +forfd 4.22) a =. e ε μ)/kt ± 1 for BE The total eergy is writte as 4.23) E = g ε e ε μ)/kt ± 1. As i the classical particle system case, with the assumptio 4.6), by differetiatig E with respect to ε, we obtai that a ε ε μ) 4.24) g kt 2 By 4.22), we have 4.25) e ε μ kt = g ± a a e ε μ kt T = a ε g kt e ε μ kt 1. +forfd for BE We ifer the from 4.24) ad 4.25) that 4.26) T = T ε μ g kt 2 g ± a εε μ). Here T = T ε ad ε = ε. The solutio of 4.26) is 4.27) ktε )= 1 ± a ) ) ε +forfd. g C +lε for BE The as i the case for classical particle systems, we derive the followig temperature formula: 4.28) kt = 1 ± a ) ) a ε +forfd. g N1 + β l ε ) for BE 4.2. Physical meaig of the temperature formula. Equatio 4.2) eables us to have a better uderstadig o the essece of temperature. I short, 1) the essece of temperature T is weighted) average eergy level, 2) the temperature T is a fuctio of distributios {a } ad the eergy levels {ε }, ad 3) the parameters {β } i the temperature formula reflects the property of the material. ).

29 28 MA AND WANG We ow discuss some further physical implicatios of the temperature formula. 1) Absolute zero for Fermi particle systems. For a Fermi particle thermodyamic system, 4.29) T =0K either a = g or a =0. Basic quatum mechaics shows that if the lower eergy level is ot fully occupied, the particles o the higher eergy level are ot stable, ad will spotaeously jump to lower eergy levels, uless there are always photos that excite the particles o the higher eergy level. Cosequetly, 4.29) ca be rewritte as { a = g for =1,,m, 4.30) T =0K a =0 for>m. This is a exact solutio of the temperature formula for Fermi particle systems. Solid state systems at T = 0 Kelvi are usually Fermi systems, sice i a solid state system, atoms ad molecules are fixed at lattice poits, ad the correspodig eergy levels are determied by the orbitig electros. Therefore such systems ca be regarded as Fermi systems cosistig of orbitig ad free electros. 2) Absolute zero for Bose particle systems. For a Bose particle system, we have 4.31) T =0K ε 1 =0,a 1 = N ad a =0 >1. This correspods exactly to the Bose-Eistei codesatio. With temperature at absolute zero, states i a Bose particle system ca oly be i two forms: a gaseous state or a codesed state of a subsystem i a object. 3) Classical systems. For a classical particle system with temperature at absolute zero, we have 4.32) T =0K a 1 = N ad a =0 >1. Here we do ot eed to assume ε 1 = 0, which correspods to supercoductivity or codesatio states of superfluids. The above results derived from the temperature formula are i agreemet with physical facts o T =0K. 4) Existece of highest temperature. Based o the theory of eergy levels i Sectio 3.4, the umber of eergy levels of all particles are fiite. Cosequetly, we ifer from the temperature formula 4.2) the

30 STATISTICAL THEORY OF HEAT 29 upper limit of T : def ε 4.33) kt max <ε max, ε max = max. 1+β l ε We used the temperature formula for classical particle systems to derive 4.33), sice at high temperature, the system ca be regarded as a classical particle system. 5. Theory of Etropy 5.1. Physical meaig of etropy. As the electros i the system represet i a atural way all the particles i the system, the eergy level theory of temperature amouts to sayig that 5.1) temperature T = average eergy level of electros. We ow develop the theory of etropy based o the dual relatio 4.1). I view of both 4.1) ad 5.1), we deduce the followig ew descriptio of etropy: 5.2) etropy = certai sese of umber of photos i the system. This equivalece 5.2) provides a startig poit for the ew theory of etropy, which we shall explore i this sectio Physical supports of etropy as umber of photos. 1) The first law of thermodyamics amouts to sayig that for a give thermodyamical system, the iteral eergy cosists of thermal eergy, mechaical eergy, iteractio eergy, etc, which ca trasform amog each other ad from oe system to aother, maitaiig the total iteral eergy ivariat. I particular we have 5.3) SdT + TdS =0, which shows that i a isolated system, thermal fluctuatio follows the rule that temperature icreasig or decreasig correspods to etropy decreasig or icreasig. At the same time, it is clear that a particle absorbs photos if ad oly if its eergy level icreases ad the umber of photos betwee particles i the system decreases, ad 5.4) a particle emits photos if ad oly if its eergy level decreases ad the umber of photos betwee particles i the system icreases. It is clear that 5.3) ad 5.4) are cosistet. This shows clearly that the first law of thermodyamics offers a strog support for etropy beig the umber of photos i the system depicted i 5.2).

31 30 MA AND WANG 2) Log-rage trasfer is oe importat characteristic of thermal eergy Q = TS. The temperature T is the average eergy levels of particles, does ot possess the log-rage trasfer feature, ad ca oly be trasferred through kietic eergies of particles. Therefore, the log-rage trasfer ca oly be achieved through the etropy S. O the other had, it is clear that photos radiatio is the oly possible cadidate. Hece 5.2) should be valid, ad i other words, the characteristic of log-rage trasfer of thermal eergy provides a physical support for 5.2). 3) First we call have the followig law of etropy trasfer. Law of Etropy Trasfer 5.1. Assume the trasfer of thermal ad other forms eergies is egligible. Whe two thermodyamic systems udergo thermal exchage, the etropy icreasig for oe system always leads to the etropy decreasig for the other system. Also, the etropy icreases for the heat iput system, ad decreases for the heat output system. This law supports 5.2). Without particle exchage, thermal eergy ca oly be trasferred through either thermal radiatio or trasfer of kietic eergy of the system particles. With thermal radiatio, eergy levels of particles i heat output system decreases. It is clear the that with photo umbers ad eergy levels i equilibrium, decreasig of eergy levels leads to the absorptio of more photos, reducig the umber of photos. For heat iput system, the kietic eergy ad eergy levels of particles icrease. This icrease of eergy levels causes emissio of more photos, for photo umbers ad eergy levels to returig to their origial equilibrium. This verifies the agreemet betwee the law of etropy trasfer ad the etropy theory 5.2) Photo umber formula of etropy. For a give thermodyamic system, i view of 5.2), we characterize etropy as the umber of photos i the photo clouds betwee system particles, or the photo desity of the photo gas i the system. As discussed earlier, thermal radiatio is simply photo radiatio γ radiatio). Also, photos are Bosos ad obey the Bose-Eistei distributio. I this case, sice the total umber of photos is ot fixed, the chemical potetial μ =0. The the BE distributio is writte as 5.5) a = g e ε μ)/kt 1.

32 STATISTICAL THEORY OF HEAT 31 Hece the total eergy of the photo gas i the system is give by 5.6) E = a ε. The correspodig partitio fuctio Z is give by 5.7) Z B = [1 e ε/kt ] g, l Z B = Cosequetly, by the etropy formula 2.18): [ S = k l Z B β ] β l Z B, we obtai that 5.8) S = k [ g l By the distributio 5.5), we have e ε/kt e ε/kt 1 + e ε/kt e ε/kt 1 =1+a g, g l[1 e ε/kt ] 1. ] g e ε/kt 1 ε. kt by which we ifer from 5.8) that 5.9) S = k [ g l 1+ a ) + a ] ε. g kt Sice for ay photo gaseous system, we always have a g, which implies that ) l 1+ a a. g g Therefore, we derive from 5.9) the followig photo umber formula of etropy: 5.10) [ S = kn kt ε a N 0 where N 0 = a is the total umber of photos, ad ε a kt represets the umber of photos i the sese of average eergy level. Notice that N 0 accouts oly the photos betwee systems particles, ot those i the clouds of electros. Basically, thaks to the mechaism of photo radiatio ad absorptio mechaism, at the equilibrium of absorptio ad radiatio, the average eergy level of the system maitais uchaged, ad represets the temperature of the system; at the same time, the umber desity) of photos i the sea of photos represets the etropy etropy desity) of the system. ],

33 32 MA AND WANG 5.3. Law of Temperature. By the temperature formula ad the etropy formula, we arrive immediately the followig law of temperature. Theorem 5.2 Law of temperature). The followig physical assertios hold true for temperature: 1) There are miimum ad maximum values of temperature with T mi =0ad T max beig give by 4.33); 2) Whe the umber of photos i the system is zero, the temperature is at absolute zero; amely, the absece of photos i the system is the physical reaso causig absolute zero temperature; 3) Nerst Theorem) With temperature at absolute zero, the etropy of the system is zero; 4) With temperature at absolute zero, all particles fills all lowest eergy levels. 6. Nature of Heat The theory of temperature ad etropy developed i the previous sectios provide a theoretical foudatio for the theory of heat. We further explore i this sectio the cosequece of the theory to reveal theatureofheat Thermal eergy. First, i classical thermodyamics, thermal eergy is defied as 6.1) ΔQ =ΔU ΔW, where ΔQ represets the thermal eergy absorbed by the system, ΔU is the chage of iteral eergy, ad ΔW is the work doe by the system. At a thermal equilibrium, we have 6.2) du = TdS pdv. Physically, this differetial equatio ca be uderstood as follows. First, for a give thermodyamical system, the absorbed released) heat dq is give by dq = du. O the other had, dq = TdS + SdT, Therefore 6.3) du = TdS + SdT. Sice the volume of the system ca chage, the chage of system temperature correspods to the work doe by the system: 6.4) SdT = work = pdv ).