1 entropy and the laws of thermodynamics 1 1.1 Entropy 2 1.1.1Classical and Statistical Entropy 4 1.1.2Aspects of Entropy 6 1.2 The 0th Law 7 1.3The 1st Law 7 1.4The 2nd Law 8 2 the third law 11 2.1 Consequences of the Third Law 12 2.2 Helium and the Third Law 14 2.3 Non-Equilibrium Degrees of Freedom 15 2.4 The Unattainability of Absolute Zero 19 3 violations of the laws of thermodynamics 23 3.1 The Second Law 23 3.1.1Fluctuations 24 3.2 The Third Law 26 4 the boson driven refrigerator 29 4.1 Introduction of the Refrigerator Model 29 4.2 Discussion 30 4.3 Non-Equilibrium Systems and Thermodynamics 36 4.4 Objective 37 5 non-equilibrium thermodynamics 39 5.1 The Stochastic Master Equation 40 5.2 Local Detailed Balance 41 5.3 The Stochastic Laws of Thermodynamics 42 5.3.1 The First Law 42 5.3.2The Second Law 43 5.4 Tight Coupling and Efficiency 45 6 boson powered refrigeration 49 6.1 Particle Currents 49 6.2 Heat Transport and Effieciency 52 6.3 Cooling power in the Limit of Absolute Zero 56 7 numerically optimized cooling 59 7.1 Conditions for Optimized Cooling 59 7.2 The Very Low Temperature Regime 63 8 finite energy level spacing 67 8.1 Dependency on δ and ∆ 73 8.2 Maximized Cooling Power 75 8.3 Higher Order Transitions 79 VII VIII contents 8.4 Heat Capacity of the Discrete Model 80 8.5 The Unattainability of Absolute Zero 82 9 summary 85 9.1 Summary and Conclusions 85 9.2 Discussion 86 9.3 Recommendations for Further Work 87 a appendix 89 a.1 C++ Source Code for the Continuous Model 89 a.2 C++ Source Code for the Discrete Model 96
A restricted proof is given of the third law of thermodynamics. The restriction resides in the fact that the proof applies only in the limit of an infinitely large system (the ‘limit L ’). An additional restriction is that it applies only to non-interacting and structureless quantum mechanical particles whose quantum states satisfy certain requirements (the ‘class 0’ of systems). The essence of the proof depends on showing that the free energy of the system, as averaged over a grand canonical ensemble and expressed per unit volume of the system, is of the form a-bTd (b>0) near the absolute zero.