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Statistical Thermodynamics. Statistical thermodynamics provides the link between the microscopic (i.e., molecular) properties of matter and its macroscopic (i.e., bulk) properties. It provides a means of calculating thermodynamic properties from the statistical relationship between temperature and energy.
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Thermodynamics is the study of how heat moves around in ‘macroscopic’ objects. Through-out these lectures, we will talk a lot about laws and models. Models are a simplified, empirical description of a real system which generally develops overtime as our knowledge progresses.
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Introduction to Statistical Thermodynamics. Henri GODFRIN. CNRS Institut Néel Grenoble. http://neel.cnrs.fr/ . Josiah Willard Gibbs worked on statistical mechanics, laying a foundation and providing a mathematical framework for quantum theory and for Maxwell's theories.
In this chapter, we will focus on two topics: (a) foundation of statistical mechanics and (b) application to isolated systems. So far, we have only calculated the entropy changes but never the absolute value.
- 192KB
- 14
- thermodynamics
- S mi-
- S statistical mechanics
- p(x) probability density “function”
- S p(x)dx =
- dμ
- N x n α
- 2 ∫ dk ̃p(k)eikx
- T t x =
- Ising model:
- Jh ik
- f = SV S−1 Z ⟨σiσj⟩ V d
- i j
- f Z = SV S−1
- D + m2)−1
- Central Limit Theorem
- Definition N
- information entropy
- WE energy surface
- P (t), Q(t) dense
- arbitrarily close Poincaré recurrence
- B ∩ Bk = ∅ ∀k ∈ N. 0
- and Density Matrices)
- time evolution
- Definition: ρA reduced density matrix
- entanglement entropy S
- 3.1. Boltzmann Equation in Classical Mechanics
- BBGKY hierarchy B B
- S⃗pS=S⃗ p′S p ⃗
- H-theorem
- Si⟩
- D +1 = O(e− N)
- V N
- E S W
- S(E {Iα})
- chemical potential μ
- S concave
- i vi
- E E
- D and , , (4.38)
- distinguishable
- indistinguishable
- Ni i
- N S
- E = EA
- NA, NB NB
- H = HA + HB + HAB ́ ̧¶ ́ ̧¶ ́ ̧¶ , (4.53)
- A B
- SB nB
- V H
- ∂A ∂H − βA
- equipartition equidistribution law
- P P
- E = EA EB
- N NA total = + NB
- NB,VB
- Gibbs free
- W μ μ
- Y N T β
- 4.5. Summary of different equilibrium ensembles
- 4.6. Approximation methods
- ⃗xσ
- . . . iN⟩ HSi⟩ = EiSi⟩
- P+P = P − −P = + 0.
- c c 1,
- grand canonical ensemble N
- occupation num-
- n = N
- 5.3. Spin Degeneracy
- N ≡ ⟨ N⟩ ˆ
- S = O(1)
- 5.5. Degenerate Bose Gas
- KMS condition K M S
- Bose-condensate superfluid
- 6. The Laws of Thermodynamics
- X Xn
- ideal gas
- X0 X X
- = α p α
- R3. fi
- S A
- A ∂E V E = E(V ) E = ( )
- Q δQ
- P V
- TS
- d E)
- = (TH − TC)(SII − SI)
- C (T
- W Q
- Din = S T S
- = ∮ S = ∫C+ S
- DS S
- W A C
- Diesel engine
- DQ
- Din
- 1 tr ⎬ T H ⎪
- T V
- 1 tr(H − N)e− ˆ μ ⎪
- N = ⟨ N⟩ ˆ
- T free
- GF
- E − DE E
- T ci
- h rihi
- T P
- S linear S(X) = ξ ⋅ X X ∈ C ξ = C C positive
- N(1) N(2) N
- P A P B
- Example: Time evolution of a population of bacteria
- = ∑Xijpeq
- v X
- N C
- C t τ0
- C i
- Metropolis algorithm τ
- E(C) C F
- C′ i C′
- C TC′
- E(C′) ≥
- m N
- Acknowledgements
As the name suggests, historically developed as an attempt to un-
to the croscopic W states of the system (counted by (E) = number of accessible states of energy E ). His proposal to relate entropy to counting problems for microscopic configurations and thereby to ideas from probability theory was entirely new and ranks as one of the major intellectual accomplishments in Physics.The systematic understanding of th...
is the subject of . That subject nowadays goes well beyond the original goal of understanding the phenomenon of heat but is more broadly aimed at the analysis of systems with a large number of, typically interacting, degrees of freedom and their description in an “averaged”, or “statistical”, or “coarse grained” manner. As such, statistical mechani...
Here, is the , defined formally by: p(x)dx = “ P ((x x , + dx)) ”. p The axioms for formally imply that we should have
1, 0 ≤ p(x) ≤ ∞ . −∞ A mathematically more precise way to think about the quantity p(x)dx is provided by measure theory, i.e. we should really think p(x)dx = dμ(x) of as defining a measure and of {E} as the corresponding collection of measurable subsets. A typical case is that p is dx a smooth (or even just integrable) function on R and that is the...
e. should be integrable with respect to the probability measure = p(x)dNx .
This is the limit of the binomial distribution for → ∞ when = , are fixed
(one might use the residue theorem to evaluate this integral). Alternatively, one may start from
the decay probability in a time interval D N . If denotes the number of decays, then the probability is obtained as: p(x) = (λT )x e− λT .
(iv) The The Ising model is a probability distribution for spins on a lattice. For each lattice
where , are parameters, and where the first sum is over all lattice bonds in
log or the two-point function in the limit → of large Z (called
“thermodynamic limit”) and a large separation between and . (See exercises.)
are e.g. the “free energy density” log , or the average number of steps
and consider the corresponding Gaussian measure for {φi} ∈ V
able. This so called “ ” explains, in some sense, the empirical evidence that the random variables appearing in various applications are distributed as Gaussians. A further important quantity associated with a probability distribution is its “infor- mation entropy”, which is defined as follows:
: Let be a subset of , and let p(x) be a, say continuous, probabil- ity density. The quantity
is called . In the context of computer science, the factor kB is dropped, and the natural log is replaced by the logarithm with base 2, which is natural to use if we think of information encoded in bits (kB is merely inserted here to be consistent with the conventions in statistical physics). More or less evident generalizations exist for more gene...
is compact. We call the . Particle trajectories do not leave this surface by energy conservation. If Hamilton’s equations admit other constants of motion, then it is natural to define a corresponding surface with respect to all constants of motion. An important feature of the dynamics given by Hamilton’s equations is
be in WE and F continuous. Then the time average is equal to the ensemble average:
comes back to that point, a phenomenon called . An intuitive “proof” of this statement can be given as follows:
Then it follows that Bn ∩ Bk = ∅ ∀nk ∈ n k , N, ≠ . Now, by Liouvilles theorem we have SB0S = SB1S = . . . = SBkS = . . . , which immediately yields SWES ≥ SB0S + . . . + SBkS + . . . = ∞. WE This clearly contradicts the assumption that is compact and therefore the statement of the theorem has to be true. Historically, the recurrence argument playe...
Quantum mechanical systems are of an intrinsically probabilistic nature, so the language of probability theory is, in this sense, not just optional but actually essential. In fact, Y⟩ really means that, if to say that the system is in a state is a self adjoint operator and
According to the rules of quantum mechanics, the of a state is de- ̈ scribed by Schrodinger’s equation i̷hd dt S Y(t)⟩ = HS Y(t)⟩ ⇒ i̷hdρ(t) = H ρ(t) ≡ Hρ(t) − ρ(t)H dt , .
The operator is called of subsystem A, and ρB that of subsystem B. The reduced density matrix reflects the limited information of an observer only having access to a subsystem. The quantity
is called the v.N.(ρA) = of subsystem A. One shows that Sv.N.(ρB) , so it does not matter which of the two subsystems we use to define it. Example: C2 Let HA = = HB with orthonormal {S basis ↑⟩ , S ↓⟩} for either system A or B. The orthonormal basis of H is S ↑↑⟩ then given by , S ↑↓⟩ , S ↓↑⟩ S ↓↓⟩ , .
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
These lecture notes are based on lectures given by Prof. Dr. Stefan Hollands at the University of Leipzig. The typesetting was done by Stefanie Riedel and Michael Gransee.
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This textbook provides comprehensive information on general and statistical thermodynamics. It begins with an introductory statistical mechanics course, deriving all the important formulae meticulously and explicitly, without mathematical shortcuts.
mechanics that deals with equilibrium thermodynamics behavior (e.g. pressure, entropy, free energy) is called Statistical Thermodynamics. Only certain problems in statistical thermodynamics are exactly solvable (e.g. ideal gas, Einstein crystal, Ising model in one and two dimensions, etc.).
