Entropy of the Universe

Magnitude	Calculation formula	Value
Total entropy of the visible part $S$ ${\ displaystyle S}$ $S$	${\frac {4\pi }{3}}s_{\gamma }l_{H_{0}}^{3}$ ${\ displaystyle {\ frac {4 \ pi} {3}} s _ {\ gamma} l_ {H_ {0}} ^ {3}}$ ${\ frac {4 \ pi} {3}} s _ {{\ gamma}} l _ {{H_ {0}}} ^ {3}$	$\sim 10^{88}$ ${\ displaystyle \ sim 10 ^ {88}}$ $\ sim 10 ^ {{88}}$
Specific entropy of photon gas $s_{\gamma }$ ${\ displaystyle s _ {\ gamma}}$ $s _ {{\ gamma}}$	${\frac {8\pi ^{2}}{90}}T_{0}^{3}$ ${\ displaystyle {\ frac {8 \ pi ^ {2}} {90}} T_ {0} ^ {3}}$ ${\ frac {8 \ pi ^ {2}} {90}} T_ {0} ^ {3}$	$\approx 1.510^{3}$ ${\ displaystyle \ approx 1.510 ^ {3}}$ $\ approx 1.510 ^ {3}$ cm ^-3

The entropy of the Universe is a quantity characterizing the degree of disorder and the thermal state of the Universe . The classical definition of entropy and the way it is calculated are not suitable for the Universe, since gravitational forces act in it, and matter itself does not form a closed system . However, it is possible to prove that in the accompanying volume the total entropy is preserved.

In a relatively slowly expanding universe, the entropy in the accompanying volume is preserved, and in order of magnitude the entropy is equal to the number of photons ^[1] .

Content

The law of conservation of entropy in the universe

In general, the increment of internal energy is:

dE=TdS-pdV+\sum \limits _{i}\mu _{i}dN_{i}.

{\ displaystyle dE = TdS-pdV + \ sum \ limits _ {i} \ mu _ {i} dN_ {i}.}

We take into account that the chemical potential of particles is equal in value and opposite in sign: ^{[ clarify ]}

dE=TdS-pdV+\sum \limits _{i}\mu _{i}(dN_{i}-d{\overline {N}}_{i}).

{\ displaystyle dE = TdS-pdV + \ sum \ limits _ {i} \ mu _ {i} (dN_ {i} -d {\ overline {N}} _ {i}).}

If we consider expansion to be an equilibrium process, then the last expression can be applied to the accompanying volume ( $V\propto a^{3}$ ${\ displaystyle V \ propto a ^ {3}}$ where $a$ ${\ displaystyle a}$ - "radius" of the universe). However, in the accompanying volume the difference of particles and antiparticles persists. Given this fact, we have:

TdS=(p+\rho )dV+Vd\rho .

{\ displaystyle TdS = (p + \ rho) dV + Vd \ rho.}

But the reason for the change in volume is expansion. If now, given this circumstance, differentiate the last expression in time:

T{\frac {dS}{dt}}=a^{3}\left[3{\frac {\dot {a}}{a}}(p+\rho )+{\dot {\rho }}\right].

{\ displaystyle T {\ frac {dS} {dt}} = a ^ {3} \ left [3 {\ frac {\ dot {a}} {a}} (p + \ rho) + {\ dot {\ rho }} \ right].}

Now, if we substitute the continuity equation that is part of the Friedman equation system:

T{\frac {dS}{dt}}=0.

{\ displaystyle T {\ frac {dS} {dt}} = 0.}

The latter means that the entropy in the concurrent volume is preserved.

Notes

↑ Valery Rubakov, Boris Shtern. Sakharov and Cosmology // Troitsky Variant, No. 10 (79), May 24, 2011

Literature

Entropy of the Universe - an article from the Physical Encyclopedia
Physics of the impossible - M .: Alpina non-fiction, 2009. - 456 p. - ISBN 978-5-91671-024-3 = Trans. from English - Michio Kaku . Physics of the Impossible. New York : Doubleday , 2008, 329 p., ISBN 978-0-385-52069-0 P.38.
Linde A.D. The Inflated Universe - 1984. - V. 144 , no. 2

Links

Tuntsov A.V. Entropy of a large canonical ensemble in an expanding universe. (Neopr.) The appeal date is September 27, 2013.
Zeldovich Ya. B. Is the formation of the universe “out of nothing” possible? Astrophysical findings. Do you need a pulsating universe? (Neopr.) Astronet . The appeal date is September 27, 2013.
Sakharov A. D. Is the formation of the universe “out of nothing” possible? Epilogue (Neopr.) . Astronet . The appeal date is September 27, 2013.

[1] Valery Rubakov, Boris Shtern. Sakharov and Cosmology // Troitsky Variant, No. 10 (79), May 24, 2011