Eements of Kinetic Theory Statistica mechanics Genera description computation of macroscopic quantities Equiibrium: Detaied Baance/ Equipartition Fuctuations Diffusion Mean free path Brownian motion Diffusion against a density gradient Drift in a fied Einstein equation Baance between diffusion and drift Einstein reation Constancy of chemica potentia Other transport phenomena Heat transport Momentum transport=viscosity Johnson noise Mosty in Kitte and Kroemer Chap. 14 Phys 11 (S006) 9 Kinetic theory 1 B. Sadouet
Thermodynamic quantities Pressure cf. Kitte and Kroemer Chapter 14 p. 391 If the partices have specuar refection by the wa, the momentum transfer for a partice arriving at ange θ is pcos P = Force da Integration on anges gives p d = t da = # d 1 d cos p cos v cos n p 0 0 0 pv n( p) p dp 3 that we woud ike to compare with the energy density u = n( p)d 3 # p = 4 # n( p) p dp 0 # 0 0 ( ) p dp non reativistic pv = pressure P = u (energy density) 3 u = 3 N V # P = N # # = same pressure as thermodynamic definition = V V U,N utra reativistic pv = P = 1 3 u Phys 11 (S006) 9 Kinetic theory B. Sadouet
Detaied Baance Consider boxes fied with gas in communication through a sma aperture Each supposed in therma equiibrium Temperature T 1, T Concentration n 1, n What is gained by one box= what is ost by the other one Detaied baance argument At equiibrium: No net fux=> gains of box 1= osses of box 1 In particuar: Outgoing fux of partices = Incoming fux of partices ( ) = ( n, ) n 1, 1 For idea gas Outgoing energy fux = Incoming energy fux J( n 1, 1 ) = J( n, ) n n Ony way for idea gas 1, ( p)d 3 p = 1, Vn Q 1, n 1 = n 1 = ( ) ( ) exp # p 1, ) d 3 p ( h 3 => equaity of temperature and chemica potentia (and pressure) Equipartition of energy can be thought as equiibrium between 3 degrees of freedom each having Phys 11 (S006) 9 Kinetic theory 3 B. Sadouet 1 d.f. = 1
Fuctuations Microscopic exchanges In a system in equiibrium, exchanges sti go on at the microscopic eve They are just baanced Macroscopic quantities= sum of microscopic quantities N x mean e # U = = s p s = s s s Z But with a finite system, reative fuctuations on such sum is of the order 1/ N e.g., fuctuation on tota energy ( ) = E E = # s e E = E E Variance not entropy µ,v Phys 11 (S006) 9 Kinetic theory 4 B. Sadouet s # s s Z ( # e ( s ( s Z # s ) * U = E = og Z Computation with partition function; By substitution we can see that E = E E = # U # # og Z = # # # Simiary when there is exchange of partices N = ogz # #ogz N = N N µ = = # N µ µ
How do systems come in equiibrium? Energy transfer If gases at different temperatures are put in contact, moecues of the hotter gas have in average higher energy and transfer net energy to the ower temperature gas => temperature equiibrium. Energy transport by diffusion. Not instantaneous => therma conductivity Heat transfer equation Simiary, transfer between wa of gas encosure and gas. =>e.g., back body radiation: equiibrium between was and photons inside cavity Momentum transfer if shear between fuid voumes reated to viscosity (see ater) Partice transfer If a gas system 1 is put in contact with another gas system where the concentration of gas moecues is ower, the higher density in system 1 wi favor diffusion of moecues to system => concentration equiibrium => Diffusion equation Phys 11 (S006) 9 Kinetic theory 5 B. Sadouet
Scattering Mean Free Path Interaction cross section dz Mean free path Consider a beam of partices incident on a target Probabiity of interaction in a sab of thickness dz = Exampe: hard spheres Probabiity of interaction in interva dz Surviva probabiity = 1 n Partice enters medium at z=0. is the attenuation ength ( ) = N ( z) 1 dz N z dz # P( z dz) = P( z) 1 dz Probabiity of interaction between z, zdz # d n dz Cross section : dimension = area Phys 11 (S006) 9 Kinetic theory 6 B. Sadouet dz d = d ( The surviva probabiity varies as: ( ) ) dp dz = P z Prob ( z )dz = exp z # dz ) P z ( ) = exp z #
Diffusion: No Concentration Gradient Brownian motion Succession of scatters:consider a partice of speed v Assume isotropic scattering, no concentration gradient => Average dispacement between two scatters aong z axis z z between scatters = scos e # s ds d cos d = 0 constant concentration => does not depend on θ Average dispacement squared between two scatters aong z axis = Variance z between scatters = ( scos ) e #s ds d cos d = 3 dn Number of scatters for partices of speed v per unit time scatters dt d z => Evoution of variance with time fixed v = v Average on distribution of veocities dt 3 d z = v dt 3 = vf ( v)dv = D = d x = d y 3 f ( v)dv dt dt D = v 3 = Diffusion coefficient Phys 11 (S006) 9 Kinetic theory 7 B. Sadouet = v
Diffusion: Concentration Gradient Suppose that we have concentration gradient aong the z axis s to first order in s dn n 0 dz 1 = n 1 = 1 # 1 1 dn o o n o dz z ( = 1 # 1 1 dn o o n o dz scos) ( z => depends on θ => Probabiity of surviva aong direction θ is such that P surviva ( s ds) = P surviva ( s) 1 ds dp # surviva ( s) = P surviva ( s) # 1 1 dn o o n o dz scos ( ds P surviva s,cos ( ) = exp # 1 o s s 1 dn o n o dz cos ( ) ) ( * e# s o 1# s 1 dn o o n o dz cos ( ) => Probabiity of interaction between s and sds P interact = P surviva ( s) ds = ds s # e o 1 s 1 dn o o o n o dz cos # ( 1 1 dn o n o dz scos ( The mean dispacement aong the z axis between coisions (keeping ony first order in reative gradient) ds s z between coisions = # scos e o 1 s s ( 1 dn o * o o ) n o dz cos ( * d cos d ), Phys 11 (S006) 9 Kinetic theory 8 B. Sadouet
Diffusion: Concentration Gradient taking into account that we get s m e Each partice undergoes v/ scatters per unit time. Hence, the mean transport veocity aong the concentration gradient is Averaging on veocity distribution 0 # s o ds o = m o m z between coisions = 1 3 o ( 3 o ) 1 dn o n o dz dt d z dt = w z = v o 3 dz = o 3 v = z between coisions = o v o 3 1 n o dn o dz = D 1 dn o n o dz 1 dn o n o dz 1 dn o n o dz Phys 11 (S006) 9 Kinetic theory 9 B. Sadouet
Transport Diffusive transport => Partice fux J z = n o w z = D dn or more generay dz Fisk s aw: Opposite to gradient This is one exampe of transport: In addition to random veocity v there is a coherent transport (or drift) veocity w and net fuxes of partices Simiar transport of charged partices expains eectric mobiity, of energy expains heat conduction, of momentum expains viscosity. Conservation of the number of partices n Consider a voume V. The decrease of the number of partices inside voume has to be equa to the tota partice fux through the surface. n d 3 x = # t J d S V S but by the Divergence Theorem J d S = # J d 3 x S V This has to be fufied whatever the voume. d S Hence the partice conservation equation: Diffusion equation Repacing J by its vaue we get J = D n n t = D n wave equation n t # J = 0 J = D n 1 c A t = A Phys 11 (S006) 9 Kinetic theory 10 B. Sadouet
Drift in an Eectric Fied Charged Partices (eectrons, hoes, ions) Drift Veocity Consider an eectric fied aong the z axis. In addition to its random veocity, each dz dt E partice wi acquire a net veocity in z direction from acceeration between coisions It is advantageous therefore to work with the time δt before the next coision instead of s. If the coision time τ c =/v is constant, the probabiity of coision between δt and δt dδt is Mean dispacement between coisions: z co. = e Each partice undergoes v/ coisions per unit time => = z between coisions v = qe m => Averaging on random veocities t # c v z = qe m t z = v cos#t 1 dt d cos d ( # c v cost 1 ) v w z = qe m qe m t e t / # c # # dt # c qe m t * = qe m # c = qe mv surviva probabiity v = qe m c =acceeration x τ c Note: this stricty appies to case where coision time τ c =/v is constant. Otherwise w = qe z m 3 v 1 = qe # 3 v m ( c eff Phys 11 (S006) 9 Kinetic theory 11 B. Sadouet
Mobiity Drift in an Eectric Fied () w = µ E Constant τ c =/v µ = q m c Ohm s aw Consider a wire of ength L and sectiona area A. If the wire is thin enough L A E = V where V is the appied votage L I = naqw = na q m V c L V = L 1 A nq m c # conductivity # c I Phys 11 (S006) 9 Kinetic theory 1 B. Sadouet
Einstein Reation Constant coision time Consider the ratio for constant τ c =/v qd µ = m v c 3 = m v c c 3 # for constant c = /v D = c m = k Tµ B q = 1 / mv 3 = = k B T Genera case If the fied is ow enough for the partice to remain in therma equiibrium at temperature τ = k B T, it can be easiy shown by integrating by part the integra giving <v> that D = m# 3 v 1 3 d dv = c eff m = µ q We then sti have D = c eff m = k BT µ q Phys 11 (S006) 9 Kinetic theory 13 B. Sadouet
Constancy of the tota chemica potentia Baance between Eectric Potentia and Density Gradient Consider a charged partice in an eectric fied aong Oz=> For constant τ c =/v, this induces a drift veocity which wi increase the concentration aong w z Any density gradient wi induce a diffusion such that Inversey a gradient of charged partices wi induce an eectric fied which wi create a drift veocity These two contributions wi baance when these two veocities are opposite => at equiibrium w z = D 1 n integrating to have the potentia w z = µ E dn dz = µ q µ E = µ 1 q n qe 1 n we get # n 1 Remembering that the interna chemica potentia is we concude that the tota chemica potentia is constant dn dz 1 n dn dz = 0 dn dz z V(z) = Edz z o qv(z) og n( z) = constant µ int z ( ) = og n( z) # n Q qv (z) µ int ( z) = constant Phys 11 (S006) 9 Kinetic theory 14 B. Sadouet
Energy and Momentum Transfer Average on random directions of scatter. Energy transfer 1> E = 1 m v v = 1 m v 1 v p = m Consequence : heat conduction viscosity Momentum transfer v v v = m 1 v Phys 11 (S006) 9 Kinetic theory 15 B. Sadouet
Therma Conductivity Consider a medium in oca therma equiibrium but with a therma gradient aong z. Diffusion wi transport energy from hotter region to cooer regions: Consider a partice 1 which just has been scattered: its initia veocity is v 1 and ange θ,ϕ. At the next coision with partice after path s, it wi transfer in average 1 m v 1 v # if partice 1comes from region of temperature T 1 and partice comes from a region of temperature T. The mean energy transport aong z per coision is 3 < Average energy transfer z >= k T 1 # T # B( ) s scos e ds with T 1 T = T T #z = z z scos < Average energy transfer #z >= 3 k T B ( z scos ) s e ds d cos v Taking into account the tota number of coisions per unit time we obtain the energy fux aong z (averaged over v) is J Qz = 3 nk T B z therma conductance = 3 nk B v 3 = 3 k T 1 T B # or J Q = # T v 3 = C v 3 = CD d cos d( ) = 3 k B where C is the heat capacity per unit voume Phys 11 (S006) 9 Kinetic theory 16 B. Sadouet d T z 3
Therma Conductivity () Heat equation CT = u where u is the energy density The oca increase of temperature with time is By same argument of energy conservation => T t = 1 C # T = D# T T t = 1 u C t u t # J Q = 0 This is the diffusion equation again Phys 11 (S006) 9 Kinetic theory 17 B. Sadouet
Reation to Brownian motion Exampe of current fuctuation across a resistor Let us consider a charge moving between pates whose votages differ by V. L E Eectron transfer If the charge is moving randomy i = q L v x and i = 0 i = q # L v x at a given time t Power spectrum However for cacuation of noise through a circuit which has some frequency dependence, we need to compute the noise as a function of the frequency. The conservation of energy impies qedx = Vdq or dq dt = q E dx V dt = q L v x Note: v is the veocity the frequency < i ( ) > dv Caed the power spectrum or the spectra density of the noise. Phys 11 (S006) 9 Kinetic theory 18 B. Sadouet
Johnson Noise area A As a mode of a resistor we consider the same system as ast side with N eectrons N = nla L q E One eectron moving a ength s aong the direction produces a square puse of ength t = s v and ampitude i = qv L cos Eectron transfer Fourier transform f ( t) For each fight path between interactions sin (#t) f ( v) = e i#t f ( t)dt = e i# (t t /) i e i# (t t /) ti for sma # # The moduus of f v ( ) at ow frequency is ti and its phase is random where the factor comes from the fact for the spectra density we combine positive and negative frequencies Fourier transform i ( ) int = 0 ( ) = f v int ( ) int i = t i int Phys 11 (S006) 9 Kinetic theory 19 B. Sadouet t i tδτ