Camille Bélanger-ChampagneLuis Anchordoqui LehmanMcGill College University City University of New York
ThermodynamicsCharged Particle and StatisticalCorrelations Mechanics in Minimum Bias Events at ATLAS
• Thermodynamic● Physics motivation potentials Enthalpy • ● Minbias event and track selection Thermodynamics IV Helmholtz free energy th • ● Azimuthal correlation results 18 SeptemberFebruary 2014 26 2012,• Gibbs free energy WNPPC 2012 • Chemical● Forward-Backward potential correlation results • Landau free energy
Thursday, September 18, 14 1 THERMODYNAMIC POTENTIALS Thermodynamic systems may do work on their environments Under certain constraintsOverview ☛ work done may be bounded from above by change in appropriately defined thermodynamic potential Imagine creating thermodynamic system from scratch in thermally insulated box of volume V Work to assemble system ☛ = U W After bringing together all particles from infinity system has internal energy U System however ☛ may not be in thermal equilibrium Spontaneous processes will then occur so as to maximize system’s entropy but internal energy remains @ U dU = Q W ☛ U(S, V ) First Law Combining Main thermodynamics identity dU = TdS PdV @U @U it follows that T = and P = { @S V @V S ✓ ◆ ✓ ◆ Second Law in form Q TdS yields ☛ dU TdS PdV C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 2 FREE WORK For porcess @ fixed ☛ (S, V ) dU 0 Overview) This says spontaneous processes in system with dS = dV =0 always lead to reduction of internal energy In systems @ fixed (S, V ) spontaneous processes drive internal energy U to a minimum
Allowing for other work processes ☛ W TdS dU Work done by system under thermodynamic conditions of constant entropy is bounded above by dU and maximum W is achieved for reversible processes It is useful to define ☛ W = W PdV free which is differential work done by system other than required to change V It follows that ☛ W TdS PdV dU free CONCLUSION: for systems @ fixed ( S, V ) ☛ W dU free C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 3 THE LEGENDRE TRANSFORMATION Consider function Z =Overview Z ( x, y ) and compute differential dZ(x, y)=Xdx + Ydy (A) x, X and y, Y are (by definition) canonically conjugate pairs To replace (x, y) by ( X, Y ) as independent variables transform Z = Z ( x, y ) according to M(X, Y )=Z xX yY it follows that ☛ dM = dZ Xdx Ydy xdX ydY (B) Substituting (A) into (B) ☛ dM = xdX ydY (C) (A) and (C) give reciprocity relations: @Z @Z @M @M = X, = Y, = x, = y @x @y @X @Y
C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 4 THE LEGENDRE TRANSFORMATION II
To replace only one Overviewof the variables tansform according to
N(x, Y )=Z yY (D)
it follows that dN = dZ Ydy ydY (E)
Substituting (A) into (E) ☛ dN = Xdx ydY
with reciprocity relations: @N @N = X, = Y @x @y
C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 5 THE LEGENDRE TRANSFORMATION III Curve in a plane can Overviewbe equally represented by: pairs of coordinates (point geometry) envelope of a family of tangent lines (line geometry) Z
y
C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 6 THE LEGENDRE TRANSFORMATION IV Overview For given Z = Z ( y ) consider tangent line that goes through point (y, Z) and has slope dZ/dy = Y
If Z intercept is N ☛ equation of line is given by (D)
Z N Y = N = Z yY y 0 ) it follows that ☛ dN = dZ ydY Ydy but ☛ dZ = Ydy dN = ydY dN ) which is reciprocal relation ☛ y = dY Legendre transformation is mapping from ( y, Z ) space to ( Y,N ) space ☛ point representation of curve into tangent line representation
C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 7 ENTHALPY
Suppose ☛ we spontaneouslyOverview create system while it is thermally insulated but in constant mechanical contact with “volume bath” @ pressure P = U + PV W quantity U + PV ☛ enthalpy Legendre transformation with respect to pairs of variables P, V H = U + PV (123) so that in equilibrium ☛ dH = dU + PdV + VdP Enthalpy should be considered as a function of S and P that are its native variables ☛ dH = TdS + VdP (124)
@H @H It follows that ☛ T = and V = (125) @S @P !P !S @T @V = (126) and Maxwell relation ☛ @P @S !S !P
C.Luis B.-Champagne Anchordoqui 2 Thursday, September 18, 14 8 ENTHALPY II Overview In general we have ☛ dH TdS + VdP
In systems @ fixed (S, P ) spontaneous processes drive enthalpy H to a minimum
For general systems dH TdS W + PdV + VdP