Thermodynamics describes the physical properties of systems composed of a large number of particles. In the beginning, it was a purely phenomenological science. Its basic notions, like pressure and temperature, where defined via the way they were measured and its only justification was the successful prediction of experimental results. Understanding of why the same notions were useful for the description of different types of matter and why the dynamics of different systems had some universal properties still did not exist.
The situation began to change with the works of Joule and in particular Boltzmann, who tried to establish a link between the theory describing the microscopic dynamics, the dynamics of single constituent particles and thermodynamics, which describes the collective dynamics of a macroscopic number of particles.
While the explanation of the irreversible collective dynamics, i.e. the second law, is still a subject of current debate [23,5], the existence of thermodynamic notions can, for a large class of systems, be understood in a satisfactory manner. The key aspect here is the Thermodynamic Limit: Intensive thermodynamic quantities approach a limit value as the size of the system increases. If this limit exists, systems with a very large number of particles, of the same type have the same thermodynamic properties, irrespective of the microscopic state.
The existence of the Thermodynamic Limit justifies the usage of thermodynamic notions like temperature for very large systems. This immediately gives rise to the following question: How large do those systems need to be for a thermodynamical description to be successful?
Contents
1. Introduction
2. Motivation: A Thermal Nanoscale Experiment
3. What is Temperature?
- 3.1. Definition in Thermodynamics
3.2. Definition in Statistical Mechanics
3.3. Local Temperature
4. Outline of the Approach
5. Quantum Central Limit Theorem
- 5.1. Model and Notation
5.2. Theorem
5.3. Sketch of the proof
5.4. Discussion of the Assumptions and Possible Generalizations
5.5. Applications in Physics
5.6. Connection to Other Existing Theorems
6. Tests and Applications of the Quantum Central Limit Theorem
- 6.1. Spectral Densities
6.2. Partition Sums
6.3. Numerical Verification
6.4. Discussion and Limitations
7. General Theory for the Existence of Local Temperature
- 7.1. Model and Partition
7.2. Thermal State in the Product Basis
7.3. Conditions for Local Thermal States
8. Ising Spin Chain in a Transverse Field
- 8.1. Coupling with Constant Width ?a: Jy = 0
8.2. Fully Anisotropic Coupling: Jx = ?Jy
8.3. Isotropic Coupling: Jx = Jy
9. Harmonic Chain
10.Estimates for Real Materials
- 10.1.Silicon
10.2.Carbon
- 10.2.1. Diamond
10.2.2. Carbon Nanotube
11.Discussion of the Length Scale Results
12.Consequences for Measurements
- 12.1.Standard Temperature Measurements
12.2.Non-thermal Local Properties
12.3.Potential Experimental Tests
13.Conclusion and Outlook
14.Deutsche Zusammenfassung
- 14.1.Einleitung
14.2.Lokale Temperatur
14.3. Zentraler Grenzwertsatz für quantenmechanische Vielteilchensysteme
14.4. Allgemeine Theorie zur Existenz lokaler Temperatur
14.5.Anwendung auf konkrete Modelle
14.6.Experimentelle Relevanz
14.7.Diskussion und Ausblick
A. Proof of the Quantum Central Limit Theorem
A.1. Two Useful Lemmas
A.2. The Pointwise Convergence of the Characteristic Function
- A.2.1. Notation
A.2.2. Proof
A.3. The Convergence of the Distributions
B. Diagonalization of the Ising Chain
C. Diagonalization of the Harmonic Chain
D. Definitions and Properties of Special Functions and Operators
- D.1. Gaussian Error Functions
D.2. Spin-1 Operators
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On the Microscopic Limit for the Existence of Local Temperature
