Brillouin Paradox

Thermally noisy resistor R (dashed line, because noisy) with diode D connected in parallel

The Brillouin paradox is an apparent paradox of thermodynamics, first described by Léon Brillouin in 1950, and represents an electrical analogue to the molecular ratchet or Maxwell's demon . The system investigated by Brillouin consists of an electrical resistor and a diode connected in parallel . A naive view would show that the heat noise of the resistor from the diode is only allowed in one direction, so that an effective electrical voltage could be tapped. Such a “rectification of the heat fluctuations” would be a way to create a perpetual motion machine of the second kind, that is, a contradiction to the second law of thermodynamics . A resolution of this contradiction is possible through a more detailed analysis, the second law therefore applies without restriction.

Solution using a Fokker-Planck equation

The Fokker-Planck equation of the system over a capacitor is:

{\ displaystyle {\ frac {\ partial p (u)} {\ partial t}} = {\ frac {\ partial} {\ partial u}} \ left (\ left ({\ frac {1} {R_ {1 } C}} + {\ frac {1} {R_ {2} (u) C}} \ right) up (u) + \ left ({\ frac {k _ {\ mathrm {B}} T_ {1}} {R_ {1} C ^ {2}}} + {\ frac {k _ {\ text {B}} T_ {2}} {R_ {2} (u) C ^ {2}}} \ right) {\ frac {\ partial p (u)} {\ partial u}} \ right)}

${\ displaystyle k _ {\ text {B}}}$ is the Boltzmann constant , the capacitance of the capacitor, the ohmic resistance, the voltage-dependent resistance of the diode and the temperatures of the components. The stationary solution of this equation is: ${\ displaystyle C}$ ${\ displaystyle R_ {1}}$ ${\ displaystyle R_ {2} (u)}$ ${\ displaystyle T_ {1/2}}$

{\ displaystyle p (u) \ varpropto \ exp \ left (- \ int \ left ({\ frac {1} {R_ {1} C}} + {\ frac {1} {R_ {2} (u) C }} \ right) u / \ left ({\ frac {kT_ {1}} {R_ {1} C ^ {2}}} + {\ frac {kT_ {2}} {R_ {2} (u) C ^ {2}}} \ right) \ mathrm {d} u \ right)}

.

The resistances are canceled for the same temperatures, the symmetrical distribution preserves the second law. This also applies in particular to the case of an ideal diode. For different temperatures you actually get a mean voltage unequal to 0, which does not contradict the Second Law. ${\ displaystyle \ exp (-Cu ^ {2} / 2kT)}$

Individual evidence

^ Léon Brillouin: Can the Rectifier Become a Thermodynamical Demon? In: Physical Review . Vol. 78, No. 5, 1950, pp. 627-628, doi : 10.1103 / PhysRev.78.627.2 .
↑ IM Sokolov: On the energetics of a nonlinear system rectifying thermal fluctuations . In: Europhysics Letters (EPL) . tape 44 , no. 3 , 1998, p. 278-283 , doi : 10.1209 / epl / i1998-00470-4 .

[1] Léon Brillouin: Can the Rectifier Become a Thermodynamical Demon? In: Physical Review . Vol. 78, No. 5, 1950, pp. 627-628, doi : 10.1103 / PhysRev.78.627.2 .

[2] IM Sokolov: On the energetics of a nonlinear system rectifying thermal fluctuations . In: Europhysics Letters (EPL) . tape 44 , no. 3 , 1998, p. 278-283 , doi : 10.1209 / epl / i1998-00470-4 .