Charged Particle Correlations in Thermodynamics and Statistical Mechanics, Lecture notes of Thermodynamics

Download Charged Particle Correlations in Thermodynamics and Statistical Mechanics and more Lecture notes Thermodynamics in PDF only on Docsity! Camille Bélanger-Champagne McGill University February 26th 2012, WNPPC 2012 Charged Particle Correlations in Minimum Bias Events at ATLAS ● Physics motivation ● Minbias event and track selection ● Azimuthal correlation results ● Forward-Backward correlation results 2 October 2014 Luis Anchordoqui Lehman College City University of New York Thermodynamics and Statistical Mechanics • Equilibrium thermodynamics Thermodynamics V Kinetic Theory of Gases I • Phase transitions and phase equilibria • Basic assumptions of kinetic theory 1Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui ENTROPY MAXIMUM When 2 bodies with and are brought in thermal contactT1 T2 Total entropy in process of equilibration increases Investigating behavior of total entropy near its maximum When equilibrium is reached should attain its maximal value heat flows from hot to cold body so that temperatures equilibrate This is ☛ Second Law of thermodynamics that follows from experiment S = S1 + S2 S is subject of first part of this class 2Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui (162) (162) gives entropy decrease caused by deviation of system’s temperature dT If 2nd body is much larger than 1st one ☛ it can be considered as bath Using and dropping index for bathed system dS = CV 2T 2 (dT )2 MORE ON THERMODYNAMIC STABILITY T by a small amount from bath temperature dU1 = CV1dT1 and second fraction in (161) can be neglectedCV2 CV1 At equilibrium ☛ (159) becomesT1 = T2 = T (160) complements condition (9) of mechanical stability CV > 0 (161)dS = 1 2T 2 ✓ 1 CV1 + 1 CV2 ◆ (dU1) 2 T > 0 ☟☟ 5Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui (163) GENERAL CASE OF THERMODYNAMIC EQUILIBRIUM dS1 = 1 T1 dU1 + P1 T1 dV1 µ1 T1 dN1 Exchanging volume means there is a movable membrane between 2 bodies Resolving (139) for we obtains to first order Consider 2 systems in contact that can exchange energy, volume, mass Exchanging mass means that this membrane is penetrable by particles dS similar expression for dS2 We could include second-order terms like those in (157) Constraints lead to total entropy change (164) (165) dU1 + dU2 = 0 dV1 + dV2 = 0 dN1 + dN2 = 0 dS = 1 T1 1 T2 ! dU1 + P1 T1 P2 T2 ! dV1 µ1 T1 µ2 T2 ! dN1 so that bodies can do work on each other to find extended conditions of stability ☟ ☟ 6Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui Requiring that in (165) has three consequences: AL QUE QUIERE CELESTE...QUE LE CUESTE (i) Energy flows from hotter body to colder body (ii) Body with a higher pressure expands (iii) Particles diffuse from body with a higher chemical potential The thermodynamic equilibrium is characterized by to that with the lower (166) (167) (168) P1 = P2 (mechanical equilibrium) T1 = T2 (thermal equilibrium) µ1 = µ2 (di↵usive equilibrium) dS 0 at the expense of body with lower pressure µ 7Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui THERMODYNAMIC PHASE DIAGRAMS Typical thermodynamic phase diagram of system Solid lines delineate boundaries between distinct thermodynamic phases and thermodynamic potentials are singular Along these lines we have coexistence of 2 phases triple point ☛ 3 phase coexistence 2.12. PHASE TRANSITIONS AND PHASE EQUILIBRIA 71 Ttemperature p pr es su re generic substance 3He 4He (a) (b) (c) Figure 2.24: (a) Typical thermodynamic phase diagram of a single component p-V -T system, showing triple point (three phase coexistence) and critical point. (Source: Univ. of Helsinki.) Also shown: phase diagrams for 3He (b) and 4He (c). What a difference a neutron makes! (Source: Brittanica.) This may in principle be inverted to yield p = p(v, T ) or v = v(T, p) or T = T (p, v). The single constraint f(p, v, T ) on the three state variables defines a surface in {p, v, T } space. An example of such a surface is shown in Fig. 2.25, for the ideal gas. Real p-v-T surfaces are much richer than that for the ideal gas, because real systems undergo phase transitions in which thermodynamic properties are singular or discontinuous along certain curves on the p-v-T surface. An example is shown in Fig. 2.26. The high temperature isotherms resemble those of the ideal gas, but as one cools below the critical temperature Tc, the isotherms become singular. Precisely at T = Tc, the isotherm p = p(v, Tc) becomes perfectly horizontal at v = vc, which is the critical molar volume. This means that the isothermal com- pressibility, κT = − 1 v ! ∂v ∂p " T diverges at T = Tc. Below Tc, the isotherms have a flat portion, as shown in Fig. 2.28, corresponding to a two-phase region where liquid and vapor coexist. In the (p, T ) plane, sketched for H2O in Fig. 2.4 and shown for CO2 in Fig. 2.29, this liquid-vapor phase coexistence occurs along a curve, called the vaporization (or boiling) curve. The density changes discontinuously across this curve; for H2O, the liquid is approximately 1000 times denser than the vapor at atmospheric pressure. The density discontinuity vanishes at the critical point. Note that one can continuously transform between liquid and vapor phases, without encountering any phase transitions, by going around the critical point and avoiding the two-phase region. In addition to liquid-vapor coexistence, solid-liquid and solid-vapor coexistence also occur, as shown in Fig. 2.26. The triple point (Tt, pt) lies at the confluence of these three coexistence regions. For H2O, the location of the triple point and critical point are given by Tt = 273.16 K Tc = 647 K pt = 611.7 Pa = 6.037× 10−3 atm pc = 22.06 MPa = 217.7 atm 2.12.2 The Clausius-Clapeyron relation Recall that the homogeneity of E(S, V, N) guaranteed E = TS−pV +µN , from Euler’s theorem. It also guarantees a relation between the intensive variables T , p, and µ, according to eqn. 2.148. Let us define g ≡ G/ν = NAµ, the Gibbs free energy per mole. Then dg = −s dT + v dp , (2.340) P V T 10Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui Equation of state for single component system may be written as Single constraint on 3 state variables f(P, V, T ) = 0 P-V-T SURFACES This may in principle be inverted to yield P = P (V, T ) V = V (T, P ) T = T (P, V ) f(P, V, T ) 2.12. PHASE TRANSITIONS AND PHASE EQUILIBRIA 73 Figure 2.26: A p-v-T surface for a substance which contracts upon freezing. The red dot is the critical point and the red dashed line is the critical isotherm. The yellow dot is the triple point at which there is three phase coexistence of solid, liquid, and vapor. 2.12.3 Liquid-solid line in H2O Life on planet earth owes much of its existence to a peculiar property of water: the solid is less dense than the liquid along the coexistence curve. For example at T = 273.1 K and p = 1 atm, ṽwater = 1.00013 cm3/g , ṽice = 1.0907 cm3/g . (2.346) The latent heat of the transition is ℓ̃ = 333 J/g = 79.5 cal/g. Thus, ! dp dT " liq−sol = ℓ̃ T ∆ṽ = 333 J/g (273.1 K) (−9.05× 10−2 cm3/g) = −1.35× 108 dyn cm2 K = −134 atm ◦C . (2.347) The negative slope of the melting curve is invoked to explain the movement of glaciers: as glaciers slide down a rocky slope, they generate enormous pressure at obstacles12 Due to this pressure, the story goes, the melting temperature decreases, and the glacier melts around the obstacle, so it can flow past it, after which it refreezes. But it is not the case that the bottom of the glacier melts under the pressure, for consider a glacier of height h = 1 km. The pressure at the bottom is p ∼ gh/ṽ ∼ 107 Pa, which is only about 100 atmospheres. Such a pressure can produce only a small shift in the melting temperature of about ∆Tmelt = −0.75◦ C. Does the Clausius-Clapeyron relation explain how we can skate on ice? My seven year old daughter has a mass of about M = 20 kg. Her ice skates have blades of width about 5 mm and length about 10 cm. Thus, even on one 12The melting curve has a negative slope at relatively low pressures, where the solid has the so-called Ih hexagonal crystal structure. At pressures above about 2500 atmospheres, the crystal structure changes, and the slope of the melting curve becomes positive. defines surface in space {P, V, T} 11Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui 72 CHAPTER 2. THERMODYNAMICS Figure 2.25: The surface p(v, T ) = RT/v corresponding to the ideal gas equation of state, and its projections onto the (p, T ), (p, v), and (T, v) planes. where s = S/ν and v = V/ν are the molar entropy and molar volume, respectively. Along a coexistence curve between phase #1 and phase #2, we must have g1 = g2, since the phases are free to exchange energy and particle number, i.e. they are in thermal and chemical equilibrium. This means dg1 = −s1 dT + v1 dp = −s2 dT + v2 dp = dg2 . (2.341) Therefore, along the coexistence curve we must have ! dp dT " coex = s2 − s1 v2 − v1 = ℓ T ∆v , (2.342) where ℓ ≡ T ∆s = T (s2 − s1) (2.343) is the molar latent heat of transition. A heat ℓ must be supplied in order to change from phase #1 to phase #2, even without changing p or T . If ℓ is the latent heat per mole, then we write ℓ̃ as the latent heat per gram: ℓ̃ = ℓ/M , where M is the molar mass. Along the liquid-gas coexistence curve, we typically have vgas ≫ vliquid, and assuming the vapor is ideal, we may write ∆v ≈ vgas ≈ RT/p. Thus, ! dp dT " liq−gas = ℓ T ∆v ≈ p ℓ RT 2 . (2.344) If ℓ remains constant throughout a section of the liquid-gas coexistence curve, we may integrate the above equation to get dp p = ℓ R dT T 2 =⇒ p(T ) = p(T0) eℓ/RT0 e−ℓ/RT . (2.345) P-V-T SURFACE OF IDEAL GAS Surface corresponding to ideal gas equation of state and its projections onto planes P (v, T ) = RT/v (P, T ), (P, v), (T, v) 12Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui 2.12. PHASE TRANSITIONS AND PHASE EQUILIBRIA 71 Ttemperature p pr es su re generic substance 3He 4He (a) (b) (c) Figure 2.24: (a) Typical thermodynamic phase diagram of a single component p-V -T system, showing triple point (three phase coexistence) and critical point. (Source: Univ. of Helsinki.) Also shown: phase diagrams for 3He (b) and 4He (c). What a difference a neutron makes! (Source: Brittanica.) This may in principle be inverted to y eld = p(v, T ) or v = v(T, p) or T = T (p, v). The single constraint f(p, v, T ) on the three state variables defines a surface in {p, v, T } space. An example of such a surface is shown in Fig. 2.25, for the ideal gas. Real p-v-T surfaces are much richer than that for the ideal gas, because real systems undergo phase transitions in which thermodynamic properties are singular or discontinuous along certain curves on the p-v-T surface. An example is shown in Fig. 2.26. The high temperature isotherms resemble those of the ideal gas, but as one cools below the critical temperature Tc, the isotherms become singular. Precisely at T = Tc, the isotherm p = p(v, Tc) becomes perfectly horizontal at v = vc, which is the critical molar volume. This means that the isothermal com- pressibility, κT = − 1 v ! ∂v ∂p " T diverges at T = Tc. Below Tc, the isotherms have a flat portion, as shown in Fig. 2.28, corresponding to a two-phase region where liquid and vapor coexist. In the (p, T ) plane, sketched for H2O in Fig. 2.4 and shown for CO2 in Fig. 2.29, this liquid-vapor phase coexistence occurs along a curve, called the vaporization (or boiling) curve. The density changes discontinuously across this curve; for H2O, the liquid is approximately 1000 times denser than the vapor at atmospheric pressure. The density discontinuity vanishes at the critical point. Note that one can continuously transform between liquid and vapor phases, without encountering any phase transitions, by going around the critical point and avoiding the two-phase region. In addition to liquid-vapor coexistence, solid-liquid and solid-vapor coexistence also occur, as shown in Fig. 2.26. The triple point (Tt, pt) lies at the confluence of these three coexistence regions. For H2O, the location of the triple point and critical point are given by Tt = 273.16 K Tc = 647 K pt = 611.7 Pa = 6.037× 10−3 atm pc = 22.06 MPa = 217.7 atm 2.12.2 The Clausius-Clapeyron relation Recall that the homogeneity of E(S, V, N) guaranteed E = TS−pV +µN , from Euler’s theorem. It also guarantees a relation between the intensive variables T , p, and µ, according to eqn. 2.148. Let us define g ≡ G/ν = NAµ, the Gibbs free energy per mole. Then dg = −s dT + v dp , (2.340) WHAT A DIFFERENCE A NEUTRON MAKES! Phase diagrams for 3He and 4He As we shall learn when we study mechanical statistics this extra neutron makes all the difference because 3He is a fermion while 4He is a boson (2p + 1n + 2 ) in 3He versus (2p + 2n + 2e) in 4He Only difference between these two atoms is a neutron: 15Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui SECOND ORDER PHASE TRANSITIONS ⌘ ⌘ = 0 ⌘ _ (Tc T ) > 0 Phases are described by order parameter that is zero in one of phases and nonzero in other phase Most of second-order transitions are controlled by temperature High-temperature (symmetric) phase ☛ For T < Tc ☛ with For chemical potential in form there are boundaries between regions with different values of µ(⌘) which are associated to different values of ⌘ µ Particles migrate from phase with higher to that with lower spatial boundary between phases moves to reach equilibrium state µ µ Since can change continuously it can adjust in uniform way without any phase boundaries decreasing its chemical potential everywhere ⌘ 16Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui SMART MATERIALS Tetragonal phase expands more rapidly in 2 directions than the 3rd one becomes cubic phase that expands uniformly in 3 directions as is raised There is no rearrangement of atoms at transition temperature Ferromagnetic ordering below the Curie point T 17Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui CHARACTERISTIC LENGTHS OF GAS Concentration of molecules is defined byn n ⌘ N V r0 = 1 n 1 3 Characteristic distance between molecules can be estimated as volume of container total number of molecules There are also long-range attractive forces between molecules Let be radius of molecule ☛ assumption (2) requires r0 a a ⌧ r0 (1) (2) but they are weak and do not essentially deviate molecular trajectories if temperature is high enough and gas is ideal 20Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui MEAN FREE PATH molecule under consideration will hit (on average) other molecules mean free path ☛ typical distance travel by molecules before colliding l = ⇡(2a)2l Volume of this cylinder ☛ volume per molecule l 1/n l = 1 n ⇠ 1 a2n = ✓ r0 a ◆2 r0 r0 a (3) Considering other molecules as non-moving ☛ that are within cylinder of height and cross-section 21Thursday, October 2, 14 C. B.-Champagne 2 Overview Luis Anchordoqui VELOCITY DISTRIBUTION FUNCTIONS Distribution of molecules in space is practically uniform, Introduce the distribution function via (v x , v y , v z ) G (v x , v y , v z ) number of molecules with velocities within elementary volume (4)dN = NG(v x , v y , v z ) dv x dv y dv z dv x dv y dv z ⌘ d3v ⌘ dv (5) around velocity vector specified by its components Integration over the whole velocity space gives total number of molecules (v x , v y , v z ) ☛ satisfies normalization conditionG (v x , v y , v z ) N (6) Distribution in space of velocities is nontrivial ☟ ☟ ☛ 1 = Z +1 1 Z +1 1 Z +1 1 dv x dv y dv z G(v x , v y , v z ) ☟ 22Thursday, October 2, 14