Classical systems in equilibrium

Transcription

1 35 Classical systes in equilibriu Ideal gas Distinguishable particles Here we assue that every particle can be labeled by an index i... and distinguished fro any other particle by its label if not by any easurable property. This is a natural assuption in classical physics, but leads to the so called Gibbs paradox of ixing entropy and also breaks down in quantu echanics. As an exaple, the icrostates of two particles Ω r, r and Ω r, r are dierent if the particles are distinguishable, or identical if the particles are indistinguishable. Particles of an ideal gas do not interact with one another. Therefore, the position and otion of a particle is not inuenced in any way by any other particle: the particles are independent and uncorrelated. Since we cannot keep track of the positions and oenta of individual particles, we can at least regard then as independent rando variables. The probability of an entire icrostate Ω is equal to the product of single-particle probabilities: fωdω f i r i, p i d 3 r i d 3 p i i Here, f i r i, p i is the PDF for only the i th particle. This eans that f i r i, p i d 3 r i d 3 p i is the probability that the i th particle would be detected within a d 3 r i volue around the position r i, oving with a d 3 p i uncertainty about the oentu p i. We will assue that all particles are identical, so that there is no reason for the functions f i to be dierent. Once we cancel out the coon factors of dω, we obtain: fω f r i, p i In canonical enseble, i fω Z e HΩ The total energy of the ideal gas does not have any interaction potential: H i Clearly we can identify: p i + Ur i fω Z i f r i, p i ] p [ Z exp i + Ur i [ ] p exp i + Ur i Furtherore, if there is no external potential or it can be neglected, then the position dependence drops out: f r, p f p Z e p / The partition function of a unifor ideal gas [ ] π Z is obtained fro the PDF noralization: d 3 r d 3 p f r, p d 3 p f p i d 3 p e p / Z dp x dp y dp z e p x +p y +p z / Z

2 36 Z Z dp e p / [ ] 3 π 3 Z dξ e ξ using the forula for Gaussian integration. Therefore, the noralized single-particle PDF is: f r, p f p e p / π Equation of state: Having the single-particle PDF, we can derive the relationship between the pressure p and other state variables for a unifor ideal gas at teperature T. This is the equation of state. The gas exerts pressure on its container because particles randoly bounce o the container walls. We will assue that all collisions with the wall are perfectly elastic, so the particles never transfer energy to the wall. This aounts to assuing the icrocanonical enseble, but in the end all ensebles are equivalent with respect to the single-particle statistics. If a particle approaches a vertical wall lying in the yz plane with velocity v v x, v y, v z and bounces o elastically, it will end up having velocity v v x, v y, v z after the collision. By conservation of oentu, the particle transfers oentu v x to the wall. Consider dn particles with initial velocity v that will bounce fro an area-da patch of the wall in a tie interval dt. Together, they will transfer oentu v x dn to the wall in tie dt, which on average corresponds to a force df v x dn/dt or pressure dp df/da v x dn/dtda. The PDF f deterines a relationship between dn, v, da and dt. If a particle with velocity v is to hit the wall within a tie interval dt, it ust be within noral distance dl v x dt fro the wall. Therefore, only the particles of velocity v within the volue d 3 r dadl v x dadt are counted in dn. We can thus obtain dn using the PDF: dn f r, p d 3 r d 3 p dp v x dadt dn v x dadt f r, pv x dadt d 3 p vxf r, pd 3 p ote that we had to ultiply the probability density f with d 3 r d 3 p to get a probability of nding a particle in phase space volue d 3 r d 3 p, and then also with in order to get the nuber of particles in d 3 r d 3 p. The volue d 3 r d 3 p centered at r and p v is sall enough that f is constant inside of it. The pressure exerted by all particles is obtained by integrating out dp over all velocities v, with a restriction that v x > which erely halves the integral over all velocities; particles hit the wall fro only one side. p d 3 p v x f r, p d 3 p p x e p / π 4π dp p p e p / π 3 5 π 4π 3 4 π 3 3 π 8 dξ ξ 4 e ξ 3

3 37 We substituted the forula for f in the second line. The, we used the spatial isotropy in the third line to represent the integral of p x as one third of the integral of p p x + p y + p z. We siultaneously switched fro the Cartesian coordinate syste to the spherical one in the integral. The spherical angles were iediately integrated out to yield a factor of 4π. Then we changed the integration variable to ξ p / in the fourth line. Lastly, the value of the reaining integral dξ ξ 4 e ξ ξ3 e ξ + 3 dξ ξ e ξ 3 4 ξe ξ dξ e ξ 3 π 8 is obtained using two integrations by parts rst u ξ 3, dv ξe ξ dξ, and then u ξ, dv ξe ξ dξ. The nal result is the equation of state: p This is the icroscopic prediction of statistical echanics. We have seen before that therodynaic teperature T is dened in a way that utilizes an ideal gas as a theroeter, assuing that its equation of state is p k B T supported by experients. We therefore learn that: k B T In canonical enseble, internal energy E is a function of teperature. In icrocanonical enseble, it is the other way round. However, ensebles are equivalent descriptions of the sae acrostate with dierent boundary conditions. We will use canonical enseble to relate internal energy to teperature. We erely need to calculate the statistical average E H ɛ of the total energy fro the average energy ɛ of one particle: ɛ d 3 r d 3 p p f r, p d 3 r d 3 p p 4π π 4π π 3 3 k BT e p / π dp p 4 e p / 5 3 π 8 The calculation of this integral proceeds in exactly the sae steps as in the case of the equation of state. The total internal energy is: E 3 k BT It is justied epirically in therodynaics, but here we derived it icroscopically using statistical principles. Heat capacity C de 3 dt k B is independent of teperature, and nite at T because the iniu energy state is not unique. Equipartition theore: Internal energy of a classical syste is k BT per independent degree of freedo.

4 38 We already proved it for a onoatoic ideal gas. A single particle has three degrees of freedo: otion in x, y, z directions. Looking at the derivation of average single-particle energy ɛ p 3 k BT for an ideal gas, the factor p could be thought of as the su of kinetic energies for individual directions, p x + p y + p z. The contribution fro each direction to ɛ is k BT by syetry. The entire gas has 3 degrees of freedo, so its internal energy is E k BT 3. Many physical systes are ade fro independent or coupled oscillators. The Hailtonian for a one-diensional haronic oscillator p + kx appears in the exponent of the Boltzann's distribution function, so the integration over coordinates in calculations of internal energy E produces siilar outcoes as the integration over oenta. Therefore, there are eectively two degrees of freedo per single spatial diension of an oscillator that contribute each a factor of k BT to E. A solid crystal of atos can be seen as a syste of coupled haronic oscillators there are springs of stiness k between any two neighboring atos, which can be atheatically decoupled using a Fourier transfor: all these oscillators yield internal energy E k BT 3. A rigid diatoic olecule has 3+ degrees of freedo: 3 coes fro the linear otion of the whole olecule through space; coes fro rotations about the two independent axes perpendicular to the line between the two atos rotational kinetic energy is again quadratic, l L x + L y, where l is olecule length and L is angular oentu. If a olecule can vibrate its length change, then it has two ore degrees of freedo coing fro the one-diensional oscillator. So, an ideal gas of rigid diatoic olecules should have E 5 k BT, and a gas of exible diatoic olecules should have E 7 k BT. The above expectations fro classical physics are often scrabled by quantu echanics. Rotational and vibrational odes cost a nite iniu energy in quantu echanics, so they ay be inactive or therally activated at low teperatures. Siilar deviations fro classical expectations can occur in correlated systes of particles. Most notably, the ideal gas constant-volue heat capacity C E/T const is never observed in nature, it always vanishes in the liit T. Boltzann distribution f r, p / e p πk B T 3/ deterines the distribution of velocities in an ideal gas. We can isolate the PDF f v for velocity distribution alone: f v vd 3 v f r, p d 3 p such that f v vd 3 v gives us the probability that a particle would have velocity within a velocity-space volue d 3 v centered at v. Using p v we get d 3 p 3 d 3 v and: f v v e v / πk B T This function describes the distribution of velocity vectors. We can also obtain the distribution f vv of speed v v. We ust integrate out f v v over velocities v that have agnitude within v, v + dv: f vvdv d 3 v f v v 4πv dv f v v v v so that: Iportant velocities: f vv 4πv e v / πk B T The ost probable speed v of a particle is: df vv dv v v v e v / v kb T

5 39 The average speed v of a particle is: v dv vf vv πk B T 4 kb T π 8kB T π 4π dv v 3 e v / πk B T kb T 4π dξ ξ 3 e ξ The root-ean-square RMS speed v rs v is the geoetric ean velocity, directly related to the average kinetic energy per particle ɛ v : v rs ɛ v 3 k 3kB T BT The velocities v, v and v rs are clearly dierent, even though they represent the sae characteristic speed scale k B T/. This is the typical speed of particles in a gas at teperature T. For nitrogen air,.33 6 kg at roo teperature T 3 K, this speed is about v rs /s. Entropy of an ideal gas of distinguishable particles at teperature T is : [ πkb T S k B {log h + 3 where h is Planck's constant. Proof: The forula for entropy in continuu systes S dω fω log fω Ω features the eleentary phase space volue Ω that cannot contain ore than one icrostate. We ust resort to quantu echanics to estiate it. It is known in quantu echanics that the position and oentu of a particle cannot be siultaneously easured to a greater accuracy than the one liited by Heisenberg uncertainty δp x δx h, where h is Planck's constant. This applies separately to every direction in space. Hence, we estiate that the single-particle phase space volue d 3 p d 3 r h 3 cannot be seen to hold ore than one particle. For particles, Ω h 3. ow we substitute this and the canonical PDF in the forula for entropy: S k B dω h 3 Z e HΩ log Z k B log h 3 + k B [ πkb T k B log h [ πkb T k B {log h Z e HΩ dω Z e HΩ HΩ ] + E T + 3 We identied the second ter in the second line as internal energy E, and then used E 3 k BT to obtain the last line.

6 Free energy of a echanically isolated syste in theral equilibriu can be written as: F E T S k B T logz G E T S µ k B T logz where Z is partition function in the canonical enseble, and Z is partition function in the grand canonical enseble. Proof: Fro S k B Ω pω log pω k B Ω pω log Z e HΩ k B logz Ω pω + k B Ω pωhω k B logz + E T for canonical enseble, we nd: Analogous holds for grand canonical enseble. Indistinguishable particles F E T S k B T logz Gibbs paradox and ixing entropy: Consider a box separated in two parts by a barrier. Let both parts have the sae volue and the sae nuber of particles of soe gas. The gas in both partitions is in equilibriu at teperature T. At soe point the partition is reoved and the particles fro dierent partitions ix. The nal equilibriu state is particles in the volue at teperature T. The change of entropy S ix due to ixing is called ixing entropy. We use the above forula for entropy S is the nal equilibriu entropy after ixing, and S, are initial entropies of the two partitions: S ix S S + S [ πkb T k B {log [ πkb T k B log h k B log k B log h { [ + 3 πkb T k B log h ] [ ] πkb T h + 3 This is interesting because one would have expected S ix. What has really changed by the reoval of the partition? Looking fro the outside, nothing about the distribution of particles throughout the box changes when the partition is reoved. In fact, the initial state of the partitioned box looks the sae as if the partition is inserted starting fro the undivided box in equilibriu! The paradox is resolved by appreciating that the forula for entropy that we used keeps track of particle labels. One ight have labeled all particles in the left partition by a blue color, and all particles in the right partition by a red color. When the partition is reoved, the blue and red particles ix and obviously achieve a dierent equilibriu state than before. The ixing process is irreversible, so S ix >. In reality, we don't have the eans to paint and label particles. Since we cannot see the color of particles, ixing does not create a new equilibriu state. We need to patch the forula for entropy and partition function in order to reove ixing entropy. After these patches are applied, nothing changes in the expressions for internal energy, therodynaic potentials, equation of state, etc.

7 4 Partition function for indistinguishable particles is: [ Z ] π! This follows fro the denition Z Ω e HΩ in which the nuber of physically distinct icrostates Ω of particles that have the sae energy H is reduced by the factor of!. If the particles are indistinguishable, then perutations aong the cannot be detected by any experient and should not be considered distinct icrostates. ote that the corrected partition function Z and hence the icrostate probability cannot be naturally factorized into a product over individual single-particle degrees of freedo. This iplies that indistinguishable particles are fundaentally not independent. There are saddle any-body correlations aong indistinguishable particles. In reality, identical particles found in nature, such as electrons, atos, etc, are physically indistinguishable and properly described only by quantu echanics. The entioned correlations lead to phenoena such as Pauli exclusion for ferions and superuidity for bosons. Entropy of an ideal gas of indistinguishable particles at teperature T is given by the Sackur-Tetrode forula: [ πkb T S k B {log h + 5 Proof: Thinking in ters of the icrocanonical enseble, entropy is the logarith of the nuber of icrostates that correspond to a given acrostate. If particles are indistinguishable, then the nuber of physically distinct icrostates is saller by the factor of! then what we coputed before for distinguishable particles. This aounts to subtracting k B log! fro the previously calculated entropy. Since is large, we ay use Stirling approxiation: log! log + Olog Then: [ πkb T S k B {log k B {log [ h πkb T h + 3 k B log! Olog yields Sackur-Tetrode forula. Gibbs paradox of ixing entropy is xed: S ix S S + S [ πkb T k B {log h [ πkb T k B log h / k B log / { [ + 5 k B log ] [ ] πkb T h πkb T h + 5

8 4 Two-level syste A two-level syste is the siplest discrete syste of constituents that can be in only two dierent states: unoccupied n i with energy and occupied n i with energy ɛ. The icrostate Ω n, n... n is specied by the occupation nubers n i {, } of all constituents. The constituents do not interact with one another, so the Hailtonian total energy is: HΩ ɛ This syste can odel ipurity atos ebedded in a crystal lattice of the host aterial. ipurity traps an electron is occupied, it has energy ɛ. Canonical probability distribution pω at teperature T is a product of single-constituent probabilities p n i : pω Z e HΩ p n i p n e ɛn + e ɛ because the constituents are non-interacting and thus independent. The noralization factor is obtained by noralization, p + p. The partition function Z is the product of noralization constants for all degrees of freedo: Z + e ɛ Internal energy as a function of teperature /k B T is: e ɛ ɛ E nɛp n ɛ + e ɛ e ɛ + n Hence, the nuber of occupied states is: Heat capacity C de dt ɛ e ɛ e ɛ + k B T k B n E ɛ i n i i e ɛ + e ɛ + e ɛ + ɛ k B T k B ɛ k B T cosh ɛ k B T vanishes both in the zero and innite teperature liits, because the iniu and axiu energy states are both unique all n i or all n i. Entropy is: S k B Ω pω log pω k B n p n log p n k [ B + e ɛ log k B + e ɛ [ log + e ɛ ɛe ɛ e ɛ log + e ɛ ] + e ɛ k B log + e ɛ + E T To obtain entropy without a reference to teperature icrocanonical, substitute p E/ɛ and p p : ] S k B [p log p + p log p k B [ E ɛ log E ɛ + E ɛ log E ɛ ] If an ] e + e ɛ ɛ log + e ɛ