THE MATHEMATICS OF THE BOSE GAS AND ITS CONDENSATION

By MARK LEE

CONTENTS INTRODUCTION THE IDEAL BOSE GAS THE CONCEPT OF BOSE-EINSTEIN CONDENSATION OVERVIEW AND OUTLINE THE DILUTE BOSE GAS IN 3D UPPER BOUND LOWER BOUND THE DILUTE BOSE GAS IN 2D GENERALIZED POINCAR´E INEQUALITIES BEC AND SUPER?UIDITY FOR HOMOGENEOUS GASES BOSE-EINSTEIN CONDENSATION SUPER?UIDITY GROSS-PITAEVSKII EQUATION FOR TRAPPED BOSONS THREE DIMENSIONS TWO DIMENSIONS BEC AND SUPER?UIDITY FOR DILUTE TRAPPED GASES ONE-DIMENSIONAL BEHAVIOR OF DILUTE BOSE GASES IN TRAPS DISCUSSION OF THE RESULTS THE 1D LIMIT OF 3D GP THEORY OUTLINE OF PROOF TWO-DIMENSIONAL BEHAVIOR IN DISC-SHAPED TRAPS THE 2D LIMIT OF 3D GP THEORY SCATTERING LENGTH 10 THE CHARGED BOSE GAS, THE ONE- AND TWO-COMPONENT CASES 101 THE ONE-COMPONENT GAS 102 THE TWO-COMPONENT GAS 103 THE BOGOLIUBOV APPROXIMATION 104 THE RIGOROUS LOWER BOUNDS 105 THE RIGOROUS UPPER BOUNDS 11 BE QUANTUM PHASE TRANSITION IN AN OPTICAL LATTICE MODEL 111 INTRODUCTION 112 RE?ECTION POSITIVITY 113 PROOF OF BEC FOR SMALL ? AND T 114 ABSENCE OF BEC AND MOTT INSULATOR PHASE 115 THE NON-INTERACTING GAS 116 CONCLUSION 12 NEW DEVELOPMENTS CHAPTER AND VOLUME V = L3 WE ARE INTERESTED IN THE “THERMODYNAMIC LIMIT”, WHICH MEANS THAT WE WILL TAKE N ??AND L ??IN SUCH A WAY THAT THE DENSITY ? = N/V IS HELD ?XED THE FACT THAT THE PARTICLES ARE NON-INTERACTING MEANS THAT THE CLASSICAL ENERGY, OR HAMILTONIAN H, IS ENTIRELY KINETIC ENERGY AND THIS, IN TURN, IS H = I (11) WHERE M IS THE PARTICLE MASS AND PI IS THE MOMENTUM OF PARTICLE I THE LOWEST (OR GROUND STATE) ENERGY OF THIS CLASSICAL SYSTEM IS, OF COURSE, 0 THE THERMODYNAMIC PROPERTIES ARE DETERMINED FROM THE PARTITION FUNCTION ZN = ? LN ZN (13) CHAPTER 1 INTRODUCTION THE PRESSURE P IS P = ??F/?V = ?KBT (14) WHERE STIRLING’S APPROXIMATION, LN N! ?N LN N, HAS BEEN USED THE AVERAGE ENERGY IS E = ?? ?? LN Z = 2NKBT (15) IN QUANTUM MECHANICS, THE HAMILTONIAN IS AN OPERATOR OBTAINED BY REPLACING EACH PJ BY ?I??J, ACTING ON THE HILBERT SPACE L2(?), WITH APPROPRIATE BOUNDARY CONDITIONS THE EIGENVALUES OF P2 = ??2?(WITH ?=LAPLACIAN =?2), FOR A BOX WITH PERIODIC BOUNDARY CONDITIONS, ARE (2??)2N2/L2, WHERE N IS A VECTOR WITH INTEGER COMPONENTS IF THE STATISTICS OF THE PARTICLES IS DISREGARDED, THE PARTITION FUNCTION, WHICH IN THE QUANTUM CASE IS GIVEN BY ZN = 1 ?EXP (??(?I ?µ)), (19) WHERE ?0 ??1 ? DENOTE THE SINGLE-PARTICLE ENERGY LEVELS (GIVEN, IN THIS CASE, BY (2??)2N2/(2ML2), WITH N ?Z3) NOTE THAT IN THIS “FREE PARTICLE” BOSONIC CASE IT IS NECESSARY THAT µ < ?0 IN THE THERMODYNAMIC LIMIT ?0 ?0 11 THE IDEAL BOSE GAS Z DP EXP (?(P2/(2M) ?µ)) ?1 (111) THIS IS A MONOTONOUSLY INCREASING FUNCTION OF µ, WHICH IS BOUNDED AS µ ?0, HOWEVER, BY THE CRITICAL DENSITY ?C(?) = G3/2(1)(2??2?/M)?3/2 (112) HERE G3/2(1) = P? ?=1 ??3/2 ?2612 THAT IS, THE DENSITY SEEMS TO BE BOUNDED BY THIS VALUE THIS IS ABSURD, OF COURSE THIS PHENOMENON WAS DISCOVERED BY EINSTEIN [E], AND THE RESOLUTION IS THAT THE PARTICLES EXCEEDING THE CRITICAL NUMBER ALL GO INTO THE LOWEST ENERGY STATE IN MATHEMATICAL TERMS, THIS MEANS THAT WE HAVE TO LET µ ?0 SIMULTANEOUSLY WITH L ??TO ?X THE DENSITY AT SOME NUMBER > ?C IN THIS CASE, WE HAVE TO BE MORE CAREFUL IN REPLACING THE SUM IN (110) BY AN INTEGRAL IT TURNS OUT TO BE SU?CIENT TO SEPARATE THE CONTRIBUTION FROM THE LOWEST ENERGY LEVEL, AND APPROXIMATE THE CONTRIBUTION FROM THE REMAINING TERMS BY AN INTEGRAL THE RESULT IS THAT, FOR ? > ?C, ? = ?C(?) + ?0, (113) WHERE ?0 = LIM L?? (114) IS THE DENSITY OF THE “CONDENSATE” THE DEPENDENCE OF µ ON L IS DETERMINED BY (110), WRITING ?N?= L3? WITH ?XED ? THE PHENOMENON THAT A SINGLE PARTICLE LEVEL HAS A MACROSCOPIC OCCUPATION, IE, A NON-ZERO DENSITY IN THE THERMODYNAMIC LIMIT, IS CALLED BOSE-EINSTEIN CON- DENSATION (BEC) NOTE THAT IN THE MODEL CONSIDERED THERE IS NO CONDENSATION INTO THE EXCITED ENERGY LEVELS, AND ONE ALWAYS HAS LIM L?? EXP (?(?I ?µ)) ?1 = (115) FOR I ?1, SINCE ?I ?µ ??I ??0 = CONST L?2