Introduction to Thermodynamics and Statistical Physics phần 9

4.1. Classical Hamiltonian H = mq˙2 − mq˙2 +V (q) 2 = 2m +V (q) . (4.9) Hamilton-Jacobi equations (4.7) and (4.8) read q˙ = p (4.10) p˙ = −∂V . (4.11) The second equation, which can be rewritten as mq¨= −∂V , (4.12) expresses Newton’s second law. 4.1.3 Example L C Consider a capacitor having capacitance C connected in parallel to an inductor having inductance L. Let q be the charge stored in the capacitor. The kinetic energy in this case T = Lq˙2/2 is the energy stored in the inductor, and the potential energy V = q2/2C is the energy stored in the capacitor. The canonical conjugate momentum is given by [see Eq. (4.3)] p = Lq˙, and the Hamiltonian (4.6) is given by 2 2 H = 2L + 2C . (4.13) Hamilton-Jacobi equations (4.7) and (4.8) read q˙ = L (4.14) p˙ = −C . (4.15) The second equation, which can be rewritten as Lq¨+ q = 0 , (4.16) expresses the requirement that the voltage across the capacitor is the same as the one across the inductor. Eyal Buks Thermodynamics and Statistical Physics 129 Chapter 4. Classical Limit of Statistical Mechanics 4.2 Density Function Consider a classical system in thermal equilibrium. The density function ρ(q¯,p¯) is the probability distribution to find the system in the point (q¯,p¯). The following theorem is given without a proof. Let H(q¯,p¯) be an Hamilto-nian of a system, and assume that H has the following form X H = Aipi +V (q¯) , (4.17) i=1 where Ai are constants. Then in the classical limit, namely in the limit where Plank’s constant approaches zero h → 0, the density function is given by ρ(q¯,p¯) = N exp(−βH(q¯,p¯)) , (4.18) where N = R dq¯R dp¯ exp(−βH(q¯,p¯)) (4.19) is a normalization constant, β = 1/τ, and τ is the temperature. The notation dq¯ indicates integration over all coordinates, namely dq¯ = dq1 dq1 · ... · dqd, and similarly dp¯= dp1 dp1 · ...· dpd. Let A(q¯,p¯) be a variable which depends on the coordinates q¯ and their canonical conjugate momentum variables p¯. Using the above theorem the average value of A can be calculates as: Z Z hA(q¯,p¯)i = dq¯ dp¯A(q¯,p¯)ρ(q¯,p¯) R dq¯R dp¯A(q¯,p¯)exp(−βH(q¯,p¯)) dq¯ dp¯ exp(−βH(q¯,p¯)) (4.20) 4.2.1 Equipartition Theorem Assume that the Hamiltonian has the following form H = Biq2 +H , (4.21) where Bi is a constant and where H is independent of qi. Then the following holds ­Biq2® = τ . (4.22) Similarly, assume that the Hamiltonian has the following form Eyal Buks Thermodynamics and Statistical Physics 130 4.2. Density Function H = Aip2 +H , (4.23) where Ai is a constant and where H is independent of pi. Then the following holds ­Aip2® = τ . (4.24) To prove the theorem for the first case we use Eq. (4.20) ­ ® R dq¯R dp¯Biq2 exp(−βH(q¯,p¯)) i i dq¯ dp¯ exp(−βH(q¯,p¯)) dqi Biq2 exp −βBiq2 dqi exp(−βBiqi ) ¶ = −∂β logµrdqi exp −βBiq2 = −∂β log βBi = 2β . (4.25) The proof for the second case is similar. 4.2.2 Example Here we calculate the average energy of an harmonic oscillator using both, classical and quantum approaches. Consider a particle having mass m in a one dimensional parabolic potential given by V (q) = (1/2)kq2, where k is the spring constant. The kinetic energy is given by p2/2m, where p is the canonical momentum variable conjugate to q. The Hamiltonian is given by p2 kq2 2m 2 (4.26) In the classical limit the average energy of the system can be easily calculated using the equipartition theorem U = hHi = τ . (4.27) In the quantum treatment the system has energy levels given by Es = s~ω , where s = 0,1,2,..., and where ω = pk/m is the angular resonance fre-quency. The partition function is given by Eyal Buks Thermodynamics and Statistical Physics 131 Chapter 4. Classical Limit of Statistical Mechanics X Z = s=0 exp(−sβ~ω) = 1−exp(−β~ω) , thus the average energy U is given by ∂logZ ~ω ∂β eβ~ω −1 Using the expansion U = β−1 +O(β) , (4.28) (4.29) (4.30) one finds that in the limit of high temperatures, namely when β~ω ¿ 1, the quantum result [Eq. (4.30)] coincides with the classical limit [Eq. (4.27)]. 4.3 Nyquist Noise Here we employ the equipartition theorem in order to evaluate voltage noise acrossaresistor. Considerthe circuitshowninthefigure below, whichconsists of a capacitor having capacitance C, an inductor having inductance L, and a resistor having resistance R, all serially connected. The system is assumed to be in thermal equilibrium at temperature τ. To model the effect of thermal fluctuations we add a fictitious voltage source, which produces a random fluctuating voltage V (t). Let q(t) be the charge stored in the capacitor at time t. The classical equation of motion, which is given by q +Lq¨+Rq˙ = V (t) , (4.31) represents Kirchhoff’s voltage law. L R V(t) ~ C Fig. 4.1. Eyal Buks Thermodynamics and Statistical Physics 132 4.3. Nyquist Noise Consider a sampling of the fluctuating function q(t) in the time interval (−T/2,T/2), namely ½q(t) −T/2 < t < T/2 T 0 else . (4.32) The energy stored in the capacitor is given by q2/2C. Using the equipartition theorem one finds ­ 2® 2C = 2 , (4.33) where ­q2® is obtained by averaging q2 (t), namely Z +∞ q2 ≡ lim dt q2 (t) . (4.34) T→∞ −∞ Introducing the Fourier transform: Z ∞ qT(t) = √ dω qT(ω)e−iωt, (4.35) −∞ one finds Z +∞ Z ∞ Z ∞ q2 = lim dt √ dω q (ω)e−iωt √ dω0 q (ω0)e−iω t T→∞ −∞ −∞ −∞ ∞ ∞ +∞ = dω q (ω) dω0 q (ω0) lim dte−i ω+ω t −∞ −∞ T→∞ |−∞ {z } Z ∞ = lim dω q (ω)q (−ω) . T→∞ −∞ 2πδ(ω+ω0) (4.36) Moreover, using the fact that q(t) is real one finds ­q2® = lim 1 Z ∞ dω |q (ω)|2 . (4.37) T→∞ −∞ In terms of the power spectrum Sq(ω) of q(t), which is defined as S (ω) = lim 1 |q (ω)|2 , (4.38) T→∞ one finds Z ∞ q2 = dω Sq(ω) . (4.39) −∞ Eyal Buks Thermodynamics and Statistical Physics 133 ... - tailieumienphi.vn