Section 3: Energy and Pressure of a dilute relativistic ideal gas ----- -----1 Partition function of a nonrelativistic gas----- The partition function is in general given by: Z = Sum over r of Exp(- beta E_r) (1.1) The total partition function is the product of the partition functions from each degree of freedom: = trans. Show that the canonical partition function is given by Z (V, T) = 1 N! Fluctuations. 1 h 3 N d p N d r N exp [ H ( p N, r N) k B T] where h is Planck's constant, T is the temperature and k B is the Boltzmann constant. $$Q_{3N}=\frac{1}{(3N)!h^{3N}} \int e^{-\beta H(q,p)}d\omega,$$. For an ideal gas, the integrals over position in (7) give VN, while the integrals over momenta separate into 3N Gaussian integrals, so that, Z= VN N!h3N I3N where I= Z 1 1 e p2=2m= 2m =2: (8) This may be written as, Z= VN 3NN! (a) Show that the grand canonical potential ( T;V; ) can be written as ( T;V; ) = kT Z 1 0 d ( ) 1 + e( ); (1) where is the chemical potential, and derivatives of the partition function Z()withrespectto =1/kBT. 4. In this case. We must compute D(E) = 1 N! This would make it very dicult to obtain analytical solutions. We must compute D(E) = 1 N! b) Calculate the average internal energy U of this string as a function of temperature T, We calculated in the lecture the distribution of velocities of the molecules of an ideal gas. 4*. 10 CHAPTER 2. Applying this equation to the neutral and ionized states of hydrogen gives n2 e Translate. Thermodynamics of the Relativistic Fermi gas in D Dimensions Francisco J. Sevilla1, and Omar Pina2, 1 1 Instituto de Fsica, Universidad Nacional Autonoma de Mexico, Apdo. It first reviews the full relativistic dispersion relation for particles with non-zero (Here ultra-relativistic means that pc mc2 where m is the mass of the particle). The correct procedure for carrying out the non-relativistic and ultra-relativistic limits is presented. Note that the partition function is dimensionless. The Hamiltonian is H(q,p) = XN i=1 p2 i 2m. Table of contents: Section 1: Partition function of a nonrelativistic gas . The ideal gas partition function and the free energy are: Z ce = VN N! Consider an ultra-relativistic ideal gas of N particles obeying the energy-momentum relation E (p) = cpl, where c is the speed of light. We explore the phase transitions of the ideal relativistic neutral Bose gas confined in a cubic box, without assuming the thermodynamic limit nor continuous approximation. which is identical to the one obtained for the ideal Fermi gas. As our particlesmay be relativistic as well as non-relativistic, we haveto use a generalformula for the relation between E and p. We have E Etotal = E0 +Ek, (st.6) 13.2 Classical limit Starting from the general formulas (13.7) for P(T,) and (13.9) for n(T,), we rst investigate the classical limit (i.e. bracketleftBigg V 2 parenleftbigg k B T planckover2pi1 c parenrightbigg 3 bracketrightBigg N Hence show that an ultra-relativistic gas also obeys the familiar ideal gas law pV = Nk B T. 4. PDF Pack. ; Z 1 = V 3 th = V 2mk BT h2 3=2; where the length scale th h 2mk BT is determined by the particle mass and the temperature. For the ultrarelativistic gas, the relation between kinetic energy and a particle momentum is E cp. Show that the canonical partition function is given by Z(V,T) = 1 N! " V 2 k BT ~c 3# N Hence show that an ultra-relativistic gas also obeys the familiar ideal gas law pV = Nk BT. 2.Change to 2P 0, V 0, at constant volume. (a) Find the free energy F of the gas. Full PDF Package Download Full PDF Package. We must compute D(E) = 1 N! PFIG-2. Only into translational and electronic modes! A short summary of this paper. The purpose of this study is to develop an original theory of a relativistic ideal gas and to prove the validity of the postulate of the special theory of The system is allowed to interchange particles and energy with the surround-ings. Where can we put energy into a monatomic gas? Science Advanced Physics Q&A Library Consider a classical gas of N indistinguishable non-interacting particles with ultra- relativistic energies, i.e. partition function Equation of State for an ideal QGP:!LFT predicts a phase-transition to a state of deconned nearly massless quarks and gluons!QCD becomes simple at high temperature and/or density e.g. (24) below. Astronuc said: True, but bear in mind that this has at least two components - an electron gas, will be treated relativistically at lower energies, than nuclei, which from the m p is hydrogen (i.e. Ranabir Chakrabarti. Solution (a) We start by calculating the partition function Z= L 3N N! With this result and b) Use the partition function of the monatomic ideal gas to check that this leads to the correct expression for its heat capacity. The term for any higher energy level is insignificant compared to the term for the ground state. where = h2 2mk BT 1=2 (9) is the thermal de Broglie wavelength. The equation U = 3PV/2 is also valid for the classical ideal gas, as discussed in Sect. Show that the canonical partition function is given by Z(V,T) = 1 N! " [Here ultra-relativistic means that pc mc 2 where m is the mass of the particle]. Nonextensive statistics of the classical relativistic ideal gas. q t r = i e i / k B T. which is the product of translational partition functions in the three dimensions. The appropriate ensemble to treat this many-body system is the grand canonical ensemble. Science. The spin is zero. Consider a three dimensional ideal relativistic gas of N particles. For a gas with N particles in a 3D box of volume V: a) Calculate the volume in phase space. 14.2 Classical limit The classical limit (non-degenerate Bose gas) corresponds to low particle densities and high temperatures. b) Use the partition function of the monatomic ideal gas to check that this leads to the correct expression for its heat capacity. 6. While the corresponding non-relativistic canonical partition function is essentially a one-variable function depending on a particular combination of temperature and volume, the relativistic canonical Non-relativistic Bosons. It will be less easy when we consider quantum ideal gases. 2 An ultra-relativistic gas Consider an ideal gas consisting of N particles obeying classical statis-tics. that an ideal gas has an energy per particle of (3/2)k i+1 are the partition functions of the two states. In this limit, the energy of an electron is related to its momentum by E(p) = c|p|.Consider N such electrons in a volume V. (a) At zero temperature, nd the chemical potential and the Fermi momentum pF for this gas as a function of N and V. 2.1 The Classical Partition Function For most of this section we will work in the canonical ensemble. That is, one has to know the distribution function of the particles over energies that de nes the macroscopic properties. [5 points] Quiz Problem 7. c) Use the grand canonical partition function to nd the chemical poten-tial of the gas. function u(T,n) that describes its energy density at a temperature T and at a frequency interval [n,n+dn]. Consider a classical ideal gas of N atoms con ned to a box of volume V in thermal equilibrium with a heat reservoir at an extremely high temperature T. The Hamiltonian of the system, H= XN l=1 jp l jc; where cis the speed of light, re ects the ultrarelativistic energy of Nnoninteracting particles: (a) Calculate the canonical partition function Z Students will remember that the partition function for a gas is calculated using the density of states, which is itself dependent on the dispersion relation. The quantum statistical mechanics of an ideal relativistic Bose gas of massive particles is discussed. (24.7.2) z e = g 1 e x p ( e, 1 / k T) The ground-state degeneracy, g 1, is one for most molecules. 14.1 Equation of state We consider a gas of non-interacting bosons in a volume V at temperature T and chemical potential . 3N i=1. (Here ultra-relativistic means that pc mc2 where m is the mass of the particle). For a perfect fluid in which the pressure is isotropic and normal to any surface we develop an expression for the stress-energy tensor as follows. derivatives of the partition function Z( ) with respect to = 1=k BT. The canonical ensemble partition function, Q, for a system of N identical particles each of mass m is given by. values. b) ind the entropy, pressure and equation of state. (4.10) We assume that the gas is enclosed in a region of volume V, and well do a purely classical calculation, neglecting discreteness of its quantum spectrum. 138 4.2.5 Discrete systems . [tex91] Relativistic ideal gas (canonical partition function) Consider a classical ideal gas of N atoms con ned to a box of volume V in thermal equilibrium with a heat reservoir at a very high temperature T. The Hamiltonian of the system, H= XN l=1 q m2c4 + p2 l c 2 mc2 ; re ects the relativistic kinetic energy of N noninteracting particles. The branch of physics studying non- 2. 1. where $d\omega$ denotes a volume element of the phase space. The thermodynamic functions of the system are obtained from the exact expression for the logarithm of the grand partition function. [tex76] Classical ideal gas (canonical ensemble) Consider a classical ideal gas of N atoms con ned to a box of volume V in thermal equilibrium with a heat reservoir at temperature T. The Hamiltonian of the system re ects the kinetic energy of 3Nnoninteracting degrees of freedom: H= X. atomic = trans +. 4. Say we have a relativistic fluid/gas, as we have in some astrophyical systems. Assume particles do not have internal degree of freedom. In the equation (st.4) g,k,T,,,h are all constant, and the energy E depends on the momentum p only. The approach outlined above can be used both at and o equilibrium. p. 2 i. b) Find the entropy and temperature, pressure and equation of state. Since this chapter is devoted to fermions, we shall omit in the following the subscript () that we used for the fermionic statistical quantities in the previous chapter. d) Consider an ideal gas of indistinguishable, non-relativistic, non-interacting, point particles of mass m. Explicitly compute the partition function Z(T,p,N) of this gas. 2.3 In Sections 2.3.1 and 2.3.2 the ideal gas partition function was calculated This Paper. for a gas of ultra-relativistic massless bosons, steep We start by reformu-lating the idea of a partition function in classical mechanics. Hence show that the pressure of a gas of such particles is one third of the (internal) energy density. CalculatingthePropertiesofIdealGases from the Partition Function F = kTlnZ Toc JJ II J I In the limit p mc (the ultra-relativistic case), we can drop the plus 1 and the minus 1, and we get (p) = pc. For a gas with n particles in a 3D box of volume V a) Calculate the volume in phase space. . Let us now compute D(E) for the nonrelativistic ideal gas. 49 4.2.4 Ultra-relativistic ideal gas . (4.10) We assume that the gas is enclosed in a region of volume V, and well do a purely classical calculation, neglecting discreteness of its quantum spectrum.
This gives the name statistical physics and de nes the scope of this subject. Advanced Physics. 7) Consider a gas of non-interacting particles which possess a hard core with radius r 0 (i.e. 2mk BT h2 1N 2 exp N l" 0 k BT G l= N lk BT ln N l L l 1 2 ln 2mk BT h2 " 0 k BT l= k BT ln L1 l N l + 1 2 ln 2mk BT h2 + " 0 k BT 7. It shows that this leads to some subtle changes in these properties which have profound consequences. We compute deviations from ideal gas behavior of the pressure, density, and Bose-Einstein condensation temperature of a relativistic gas of charged scalar bosons caused by the current-current interaction induced by electromagnetic quantum fluctuations treated via scalar quantum electrodynamics. From the partition function of the grand canonical ensemble, the distribution function f( ) for derivatives of the partition function Z()withrespectto =1/k B T. b) Use the partition function of the monatomic ideal gas to check that this leads to the correct expression for its heat capacity. Ideal monatomic gases. 138 4.5.1 Canonical distribution and partition function 144 4.5.2 The difference between P(En) and However, at first glance, since the expressions for the energy densities of an ultrarelativistic classical ideal gas and a non-relativistic ideal classical gas are similar, we could think that the similarity in this case does not also appear. It first reviews the full relativistic dispersion relation for particles with non-zero 3.5. . If g = 3/2 then gamma = (g+1)/g = 5/3. Download Download PDF. Z l= L s 2N N l! To evaluate Z 1, we need to remember that energy of a molecule can be broken down into internal and external com-ponents. their kinetic energy - momentum relation is given by = pc, with c the speed of light and p the magnitude of the particle's momentum.
derivatives of the partition function Z( ) with respect to = 1=k BT. 22 February: Non-relativistic, classical, ideal gas; Energy levels in a box; Distinguishable vs. indistinguishable canonical partition functions; Internal energy and entropy for indistinguishable case. Let us now compute D(E) for the nonrelativistic ideal gas. statistical mechanics of the Fermi gas follows directly from the Grand Caronical Partition function (5.24) and the Fermi function n( ) = e( ) +1 1 (8.1) which gives the expected number of Fermions in energy state . (5-6) read: P = NT V; S = 5 2 N + Nln " V N mT 2 3=2 #; (8) E = 3 2 NT; = Tln " V N mT 2 3=2 #: (9) The most common example of a photon gas in equilibrium is the black-body radiation.. Photons are part of a family of particles known as bosons, particles that follow BoseEinstein 2.7.5 Entropy for an ideal gas 48 2.7.6 Example system . depends on its momentum, i.e. (b) Find an expression for the energy U of the gas. Classical, ultrarelativistic ideal gas is confined in twodimensional area with size LLx y. The thermal de Broglie Show that at high temperatures E = 3 Nk B T, and the equation of state coincides with that of a classical ultra-relativistic gas. Section 2: Energy and Pressure of a dilute nonrelativistic ideal gas . Hadron Production in Ultra-relativistic Nuclear Collisions: Quarkyonic Matter and a Triple Point in the Phase Diagram of QCD are resonance-dominated, the system can be replaced by an idealgas of all possible resonances [49,50]. protons). , where z is the single particle partition function in The partition function for one oscillator is Q1 D Z1 1 exp p2 2m C 1 2 m!2 0x 2 dxdp h: (3) The integrations over the Gaussian functions are precisely as in the ideal gas, so that Q1 D 1 h 2m 1=2 2 m!2 0 1=2 D kT h!N0; (4) introducing hNDh=2for convenience. No external field is applied so the gas has zero potential energy. the non-degenerate Fermi gas), which corresponds, as Ultra-relativistic fermions: Consider a non-interacting ideal gas of fermions with spin 1/2 in three dimensions. 5. The thermodynamical functions of the ideal gas from Eqs. Compare Eq. This chapter repeats the derivation of the partition function for a gas, and hence of the other thermodynamic properties that can be obtained from it, but this time includes relativistic effects. 3.Change 2P 0, 2V 0, at constant pressure. (Here the term ultra-relativistic means that [p|c mc where m is the mass of the particle.) 13.1 Equation of state Consider a gas of N non-interacting fermions, e.g., electrons, whose one-particle wave-functions r( r) are plane-waves. For a classical gas with no interactions, the Hamiltonian doesn't depend on the position, so we can immediately see that the partition function $Z\sim V^N$ and therefore $$p = \frac{\partial}{\partial V}(kT\log{Z})=\frac{NkT}{V}$$ So an ultra-relativistic gas behaves just like an ideal gas for many purposes. THERMODYNAMICS 0th law: Thermodynamic equilibrium exists and is characterized by a temperature 1st law: Energy is conserved 2nd law: Not all heat can be converted into work 3rd law: One cannot reach absolute zero temperature. is much shorter than the density variations, the gas may be thought as divided into small subsystems in which the thermodynamics of a homogenous gas can be applied, such that the following discussion is also true for trapped samples. V 2 k BT ~c 3# N Hence show that an ultra-relativistic gas also obeys the familiar ideal gas law pV = Nk BT. 2) Ultra-relativistic gas In a relativistic gas you can ignore the mass and the energy is then E = pc. When does this break down? 6.2: Consider a degenerate, ultra-relativistic (mc2 cpF) gas of non-interacting electrons, where pF is the Fermi momentum. 6. The gas is con ned within a square wall of size L. Assume that the temperature is T . mT 2 3N=2; F = NT NTln " V N mT 2 3=2 #; where we have assumed N 1 and used Stirlings formula: lnN! To obtain the result for (T), recall that the mean number of particles is given by hNi = T,V, (22) which, according to Eq. The Hamiltonian is H(q,p) = XN i=1 p2 i 2m. here for the classical ideal gas because we will nd a closed form expression for (T) as a function of n, Eq. genneth. For MB particles, this is related to the single-particle partition function, Z which is also written as ) we may write the equation of state for an ideal gas as Pe = kNA Ideal gas equation of state from the Grand potential The Grand Canonical ensemble can make some calculations particularly simple. (4.10) We assume that the gas is enclosed in a region of volume V, and well do a purely classical calculation, neglecting discreteness of its quantum spectrum. Considering only thermodynamic aspects, Wien showed that such a function must obey [4], u(T,n) = n3 f(n/T), (1) where f was an unknown function; this is called now the Wien displacement law. Show that 3 pV = E. Show that at zero temperature pV 4 / 3 = const. 3. $$d\omega=dq_1dp_1\cdots dq_{3N}dp_{3N},$$. We obtain expressions for those quantities in the ultra-relativistic and Astrophysical Gas Dynamics: Relativistic Gases 11/73 and (17) 3.2 Stress energy tensor for a perfect fluid The above characterisation of the stress-energy tensor is valid in general. (4 V (m c h) 3 e u K 2 (u) u) N represents the ordinary partition function of a relativistic ideal gas.
This gives the name statistical physics and de nes the scope of this subject. Advanced Physics. 7) Consider a gas of non-interacting particles which possess a hard core with radius r 0 (i.e. 2mk BT h2 1N 2 exp N l" 0 k BT G l= N lk BT ln N l L l 1 2 ln 2mk BT h2 " 0 k BT l= k BT ln L1 l N l + 1 2 ln 2mk BT h2 + " 0 k BT 7. It shows that this leads to some subtle changes in these properties which have profound consequences. We compute deviations from ideal gas behavior of the pressure, density, and Bose-Einstein condensation temperature of a relativistic gas of charged scalar bosons caused by the current-current interaction induced by electromagnetic quantum fluctuations treated via scalar quantum electrodynamics. From the partition function of the grand canonical ensemble, the distribution function f( ) for derivatives of the partition function Z()withrespectto =1/k B T. b) Use the partition function of the monatomic ideal gas to check that this leads to the correct expression for its heat capacity. Ideal monatomic gases. 138 4.5.1 Canonical distribution and partition function 144 4.5.2 The difference between P(En) and However, at first glance, since the expressions for the energy densities of an ultrarelativistic classical ideal gas and a non-relativistic ideal classical gas are similar, we could think that the similarity in this case does not also appear. It first reviews the full relativistic dispersion relation for particles with non-zero 3.5. . If g = 3/2 then gamma = (g+1)/g = 5/3. Download Download PDF. Z l= L s 2N N l! To evaluate Z 1, we need to remember that energy of a molecule can be broken down into internal and external com-ponents. their kinetic energy - momentum relation is given by = pc, with c the speed of light and p the magnitude of the particle's momentum.
derivatives of the partition function Z( ) with respect to = 1=k BT. 22 February: Non-relativistic, classical, ideal gas; Energy levels in a box; Distinguishable vs. indistinguishable canonical partition functions; Internal energy and entropy for indistinguishable case. Let us now compute D(E) for the nonrelativistic ideal gas. statistical mechanics of the Fermi gas follows directly from the Grand Caronical Partition function (5.24) and the Fermi function n( ) = e( ) +1 1 (8.1) which gives the expected number of Fermions in energy state . (5-6) read: P = NT V; S = 5 2 N + Nln " V N mT 2 3=2 #; (8) E = 3 2 NT; = Tln " V N mT 2 3=2 #: (9) The most common example of a photon gas in equilibrium is the black-body radiation.. Photons are part of a family of particles known as bosons, particles that follow BoseEinstein 2.7.5 Entropy for an ideal gas 48 2.7.6 Example system . depends on its momentum, i.e. (b) Find an expression for the energy U of the gas. Classical, ultrarelativistic ideal gas is confined in twodimensional area with size LLx y. The thermal de Broglie Show that at high temperatures E = 3 Nk B T, and the equation of state coincides with that of a classical ultra-relativistic gas. Section 2: Energy and Pressure of a dilute nonrelativistic ideal gas . Hadron Production in Ultra-relativistic Nuclear Collisions: Quarkyonic Matter and a Triple Point in the Phase Diagram of QCD are resonance-dominated, the system can be replaced by an idealgas of all possible resonances [49,50]. protons). , where z is the single particle partition function in The partition function for one oscillator is Q1 D Z1 1 exp p2 2m C 1 2 m!2 0x 2 dxdp h: (3) The integrations over the Gaussian functions are precisely as in the ideal gas, so that Q1 D 1 h 2m 1=2 2 m!2 0 1=2 D kT h!N0; (4) introducing hNDh=2for convenience. No external field is applied so the gas has zero potential energy. the non-degenerate Fermi gas), which corresponds, as Ultra-relativistic fermions: Consider a non-interacting ideal gas of fermions with spin 1/2 in three dimensions. 5. The thermodynamical functions of the ideal gas from Eqs. Compare Eq. This chapter repeats the derivation of the partition function for a gas, and hence of the other thermodynamic properties that can be obtained from it, but this time includes relativistic effects. 3.Change 2P 0, 2V 0, at constant pressure. (Here the term ultra-relativistic means that [p|c mc where m is the mass of the particle.) 13.1 Equation of state Consider a gas of N non-interacting fermions, e.g., electrons, whose one-particle wave-functions r( r) are plane-waves. For a classical gas with no interactions, the Hamiltonian doesn't depend on the position, so we can immediately see that the partition function $Z\sim V^N$ and therefore $$p = \frac{\partial}{\partial V}(kT\log{Z})=\frac{NkT}{V}$$ So an ultra-relativistic gas behaves just like an ideal gas for many purposes. THERMODYNAMICS 0th law: Thermodynamic equilibrium exists and is characterized by a temperature 1st law: Energy is conserved 2nd law: Not all heat can be converted into work 3rd law: One cannot reach absolute zero temperature. is much shorter than the density variations, the gas may be thought as divided into small subsystems in which the thermodynamics of a homogenous gas can be applied, such that the following discussion is also true for trapped samples. V 2 k BT ~c 3# N Hence show that an ultra-relativistic gas also obeys the familiar ideal gas law pV = Nk BT. 2) Ultra-relativistic gas In a relativistic gas you can ignore the mass and the energy is then E = pc. When does this break down? 6.2: Consider a degenerate, ultra-relativistic (mc2 cpF) gas of non-interacting electrons, where pF is the Fermi momentum. 6. The gas is con ned within a square wall of size L. Assume that the temperature is T . mT 2 3N=2; F = NT NTln " V N mT 2 3=2 #; where we have assumed N 1 and used Stirlings formula: lnN! To obtain the result for (T), recall that the mean number of particles is given by hNi = T,V, (22) which, according to Eq. The Hamiltonian is H(q,p) = XN i=1 p2 i 2m. here for the classical ideal gas because we will nd a closed form expression for (T) as a function of n, Eq. genneth. For MB particles, this is related to the single-particle partition function, Z which is also written as ) we may write the equation of state for an ideal gas as Pe = kNA Ideal gas equation of state from the Grand potential The Grand Canonical ensemble can make some calculations particularly simple. (4.10) We assume that the gas is enclosed in a region of volume V, and well do a purely classical calculation, neglecting discreteness of its quantum spectrum. Considering only thermodynamic aspects, Wien showed that such a function must obey [4], u(T,n) = n3 f(n/T), (1) where f was an unknown function; this is called now the Wien displacement law. Show that 3 pV = E. Show that at zero temperature pV 4 / 3 = const. 3. $$d\omega=dq_1dp_1\cdots dq_{3N}dp_{3N},$$. We obtain expressions for those quantities in the ultra-relativistic and Astrophysical Gas Dynamics: Relativistic Gases 11/73 and (17) 3.2 Stress energy tensor for a perfect fluid The above characterisation of the stress-energy tensor is valid in general. (4 V (m c h) 3 e u K 2 (u) u) N represents the ordinary partition function of a relativistic ideal gas.