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Help:Displaying a formula

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|}}
  • References <ref name="???">reference</ref>,<ref name="???"/>,<references/>,{{rp|p.103}}
  • References (Harvard with pages)
<ref name="Rybicki 1979 22">{{harvnb|Rybicki|Lightman|1979|p=22}}</ref>
{{Reflist|# of columns}} 
=== Bibliography ===
{{ref begin}}
*{{Cite book|etc |ref=harv}}
{{ref end}}

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Chi-squared distributions[edit]

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[edit]

Heavy tail distributions

Distribution character \alpha
Levy skew alpha-stable distribution continuous, stable 0\le \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[edit]

A Table of Statistical Mechanics Articles
Maxwell Boltzmann Bose-Einstein Fermi-Dirac
Particle Boson Fermion

Partition function
Identical particles#Statistical properties
Statistical ensemble|Microcanonical ensemble | Canonical ensemble | Grand canonical ensemble


Maxwell-Boltzmann statistics
Maxwell-Boltzmann distribution
Boltzmann distribution
Derivation of the partition function
Gibbs paradox

Bose-Einstein statistics Fermi-Dirac statistics
gas in a box
gas in a harmonic trap
Gas Ideal gas

Bose gas
Bose-Einstein condensate
Planck's law of black body radiation

Fermi gas
Fermion condensate

Classical Chemical equilibrium


Continuum mechanics[edit]

Work pages[edit]

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} \le 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.


  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.