【豪豬教授的手寫筆記】L05：HH0057-temperature

-HH0057-TemperatureHsiu-Hau Linhsiuhauphys.nthu.edu.tw(Sep 30,2012)equilibrium conditionWhen two systems are in thermal contact,the most probable configurationcorresponds to the maximum of the total multiplicity function g1g2,d(g1g2)=0!1g1g1E1N1?E1=U1=1g2g2E2N2?E2=U2(1)where U1,U2are the energies of the most probable configuration and can beviewed as average energies of the subsystems in thermal equilibrium.Theabove notation is a bit tedious and can be written in the textbook notationas well(though a bit confusing),1g1g1U1N1=1g2g2U2N2!logg1U1N1=logg2U2N2(2)The equilibrium condition motivates us to define a physical quantity:1 loggUN?fundamental temperature(3)The fundamental temperature is related to the usual temperature T by theBolzmann constant,=kBT.temperature as an energy scaleThe above definition of temperature is quite far from our common intuition.Can we improve our understanding better?1 loggUN?!1?U=?logg,(4)-HH0057-temperature2Figure 1:Temperature can be viewed as the energy scale for the exponentialgrowth of the multiplicity function g(U).Its role is pretty much as same asthe half-life time in the decaying process.where?U is small energy derivation from the average energy in equilibrium.Note that?logg=logg(U+?U)?logg(U)=logg(U+?U)/g(U).Theabove relation can be brought into the form,1?U=logg(U+?U)/g(U)?!g(U+?U)=g(U)e?U/(5)The multiplicity g(U)grows by a factor e?U/when U increase to U+?U.That is to say,sets the energy scale of the exponential growth of themultiplicity g(U).Take decaying process as a comparing example.The surviving quantityevolves as N(t)=N(0)e?t/,where is the half-life time.To pin down thecharacteristic time scale of the decaying process,it is unwise to use the decayrate dN/dt because it also depends on the initial condition N(0).A betterchoice is to use the rate of logN,?dlogNdt=?1NdNdt=1.(6)The surviving quantity N(t)decays by a factor e?t/when t increases tot+?t.Obviously,the half time sets the time scale for the decaying process.the concept of reservoirGoing back to the example of two spin systems in thermal equilibrium.Oneinteresting limit is N2?N1and the larger system is called“reservoir”.The-HH0057-temperature3multiplicity function of the reservoir isg(E2)=g(0)exp?E222N2m2B2?!logg=logg(0)?E222N2m2B2(7)According to the definition of temperature,(E)of the reservoir at energyclose to its equilibrium value E2=U2+?U is1 loggEN=?E2N2m2B2=?U2N2m2B2?UN2m2B2.(8)In the limit N2?N1,the energy exchange between the subsystems?U U2and the second term can be dropped.We arrive at the very useful approxi-mative relation,1?U2N2m2B2=12?!2=?N2U2m2B2/?1u2(9)where u2=U2/N2is the average energy of each spin in the reservoir.Twostrange behaviors need further explanations:(1)If the spin excess is positive(thus corresponds to negative energy),the temperature is positive.How-ever,when spin excess is negative,the temperature is negative as well.It isimportant to keep in mind that negative temperature is indeed possible forsome statistical systems.(2)The temperature of the reservoir is inverselyproportional to the average energy of each constituent.Therefore,you shouldabandon the intuition that the temperature is proportional to the averageenergy of each constituent.density of states for ideal gasNow we try to compute the multiplicity function of an ideal gas.From themultiplicity function,we can then derive the average energy of each molecule,u=(3/2),in thermal equilibrium.Consider a monoatomic ideal gas of Ngas molecules.The energy of the system isE=p21x2m+p21y2m+p21z2m+p2Nx2m+p2Ny2m+p2Nz2m.(10)Adding in the favor of quantum physics:the momentum p1x=n1x/L isquantized and same for all others.Introduce the 3N dimensional vector,n=(n1x,n1y,n1z,nNx,nNy,nNz),(11)-HH0057-temperature4Figure 2:A monoatomic ideal gas of N molecules.All stationary stateswith quantized momenta can be uniquely labelled by the vector n in the 3Ndimensional space and its magnitude n/pE is proportional to the squareroot of the total energy E.the total energy of the system readsU=222mL2n2,(12)where n=|n|=qn21x+n21y+n21z+n2Nx+n2Ny+n2Nz.For realisticsize of L,the energy levels are extremely close and it is better to view thediscrete energy levels as continuous.Instead of computing the multiplicityfor each energy level,we compute the total multiplicity for all levels in theenergy interval(E,E+?E),XE0g(E0)=(number of grid points in the n?space)D(E)?E(13)where the summation is over energy E0in the interval(E,E+?E)and D(E)is the density of states.The number of grid points in the thin spherical shellis roughly its volume in the 3N space.Therefore,D(E)?E C n3N?1?n,where C is some constant.Making use of the relation n=(p2mL/)E,the density of states for a monoatomic ideal gas isD(E)C n3N?1dndE=C0E(3N?1)/21pE!D(E)C0 E32N?1(14)In thermal equilibrium,average energy and temperature are related,1=logDU=1DDU=32N?11U32N 1U.(15)-HH0057-temperature5The average energy of the ideal gas is approximately U=(3N/2)in thethermodynamic limit N!1.The average ene