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$dU=\underbrace {\delta Q} _{\delta Q}+\underbrace {\delta W_{irr}+\delta W_{rev}} _{\delta W}$ $dU=\underbrace {\delta Q+\delta W_{irr}} _{T\,dS}+\underbrace {\delta W_{rev}} _{-P\,dV}$ • References {{note_label|Wood1992||}} {{ref_harvard|Wood1992|Wood, 1992|}}
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• References (Harvard with pages)
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==References==
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Chi-squared distributions

 distribution $\sigma ^{2}$ $\nu$ scale-inverse-chi-squared distribution $\sigma ^{2}$ $\nu$ inverse-chi-squared distribution 1 $1/\nu$ $\nu$ inverse-chi-squared distribution 2 $1$ $\nu$ inverse gamma distribution $\beta /\alpha$ $2\alpha$ Levy distribution $c$ $1$ Heavy tail distributions

Heavy tail distributions

 Distribution character $\alpha$ Levy skew alpha-stable distribution continuous, stable $0\leq \alpha <2$ Cauchy distribution continuous, stable $1$ Voigt distribution continuous $1$ Levy distribution continuous, stable $1/2$ scale-inverse-chi-squared distribution continuous $\nu /2>0$ inverse-chi-squared distribution continuous $\nu /2>0$ inverse gamma distribution continuous $\alpha >0$ Pareto distribution continuous $k>0$ Zipf's law discrete $s-1>-1???$ Zipf-Mandelbrot law discrete $s-1>-1???$ Zeta distribution discrete $s-1>-1???$ Student's t-distribution continuous $\nu >0$ Yule-Simon distribution discrete $\rho$ ? distribution continuous $s-1>-1???$ Log-normal distribution??? continuous $\rho$ Weibull distribution??? ? $?$ Gamma-exponential distribution??? ? $?$ Statistical Mechanics

A Table of Statistical Mechanics Articles
Maxwell Boltzmann Bose-Einstein Fermi-Dirac
Particle Boson Fermion
Statistics
Statistics Bose-Einstein statistics Fermi-Dirac statistics
Thomas-Fermi
approximation
gas in a box
gas in a harmonic trap
Gas Ideal gas
Chemical
Equilibrium
Classical Chemical equilibrium

Others:

Work pages

To fix:

 (subtract mean) (no subtract mean) Covariance Correlation Cross covariance Cross correlation see ext Autocovariance Autocorrelation Covariance matrix Correlation matrix Estimation of covariance matrices

Proof: Introduce an additional heat reservoir at an arbitrary temperature T0, as well as N cycles with the following property: the j-th such cycle operates between the T0 reservoir and the Tj reservoir, transferring energy dQj to the latter. From the above definition of temperature, the energy extracted from the T0 reservoir by the j-th cycle is

$dQ_{0,j}=T_{0}{\frac {dQ_{j}}{T_{j}}}\,\!$ Now consider one cycle of the heat engine, accompanied by one cycle of each of the smaller cycles. At the end of this process, each of the N reservoirs have zero net energy loss (since the energy extracted by the engine is replaced by the smaller cycles), and the heat engine has done an amount of work equal to the energy extracted from the T0 reservoir,

$W=\sum _{j=1}^{N}dQ_{0,j}=T_{0}\sum _{j=1}^{N}{\frac {dQ_{j}}{T_{j}}}\,\!$ If this quantity is positive, this process would be a perpetual motion machine of the second kind, which is impossible. Thus,

$\sum _{i=1}^{N}{\frac {dQ_{i}}{T_{i}}}\leq 0\,\!$ Now repeat the above argument for the reverse cycle. The result is

$\sum _{i=1}^{N}{\frac {dQ_{i}}{T_{i}}}=0\,\!$ (reversible cycles)

In mathematics, it is often desireable to express a functional relationship $f(x)\,$ as a different function, whose argument is the derivative of f , rather than x . If we let y=df/dx  be the argument of this new function, then this new function is written $f^{\star }(y)\,$ and is called the Legendre transform of the original function.

References

1. ^ Herrmann, F.; Würfel, P. (2005). "Light with nonzero chemical potential". Am. J. Phys. American Association of Physics Teachers. 78 (3): 717–721. doi:10.1119/1.1904623. Retrieved 2012-12-20. A necessary condition for Planck's law to hold is that the photon number is not conserved, implying that the chemical potential of the photons is zero. While this may be unavoidably true on very long timescales, there are many practical cases that are dealt with by assuming a nonzero chemical potential, which yields an equilibrium distribution which is not Planckian.
 Wall (Impermeable) Rigid T-Ins No IW External δQh δWx dT dS dP dV dN dV=0 δQh=0 δWx=0 Process Isobaric process - - - - 0 - 0 no - - constant pressure reservoir Isochoric process - - - - - 0 0 - yes - - Isothermal process - - 0 - - - 0 - no - constant temperature reservoir Isentropic process 0 0 - 0 - - 0 - yes yes - Adiabatic process 0 - - - - - 0 - yes - - x-Isolated System Mechanically Is. (Guha) - - - - - 0 0 yes - - - Adiabatic Is 0 - - - - - 0 - yes - - Thermally Is. 0 0 - 0 - - 0 - yes yes - Closed system - - - - - - 0 - - - - Isolated system 0 0 - 0 - 0 0 yes yes yes - Open system - - - - - - - no no no - Conserved Thermodynamic potential U: Internal energy 0 0 - 0 - 0 0 yes yes yes - F: Helmholtz free energy - - 0 - - 0 0 yes no - constant temperature reservoir H: Enthalpy 0 0 - 0 0 - 0 no yes yes constant pressure reservoir G: Gibbs free energy - - 0 - 0 - 0 no no - constant pressure and temperature reservoir

δQh is heat, δWx is irreversible work, so TdS=δQh+δWx. If both are zero, then dS=0. For the 3 possible walls, "T ins" means thermally insulated, and "No IW" means no irreversible work. "External" specifies the region external to the system.