Equation 6 · The Arrow of Time and the Engine of Evolution
What does this equation mean?
Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.
This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.
Read it piece by piece
Symbol e^-β W
β W is part of the quantity the equation computes from the expression on the right.
Symbol e^-β Δ F
β Δ F is one of the signed contributions combined to compute the quantity on the left.
Symbol β
β is part of the quantity the equation computes from the expression on the right.
Symbol k_B
occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
Symbol T
T occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →change
Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.
subscript
The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.
superscript
A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.
See an illustrated explanation →Denominator: k_B T
The complete quantity below the fraction bar; it must be nonzero for this division.
How to interpret it
With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.
What the article says around this equation
Christopher Jarzynski’s 1997 equality was the opening result. For a system beginning in thermal equilibrium and then driven away from it by an arbitrarily fast, arbitrarily far-from-equilibrium protocol, the exponential average of the work W performed on it recovers the equilibrium free-energy difference F between the protocol’s start and end points exactly: . [ 5 ] . The result does not say average work equals F — dissipation makes the ordinary average of W larger than F whenever the protocol is not quasistatic — it says a specific nonlinear average of a driven, dissipative, irreversible process reproduces an equilibrium quantity exactly, which is a…
Read the full surrounding passage
Christopher Jarzynski’s 1997 equality was the opening result. For a system beginning in thermal equilibrium and then driven away from it by an arbitrarily fast, arbitrarily far-from-equilibrium protocol, the exponential average of the work W performed on it recovers the equilibrium free-energy difference F between the protocol’s start and end points exactly: . [ 5 ] . The result does not say average work equals F — dissipation makes the ordinary average of W larger than F whenever the protocol is not quasistatic — it says a specific nonlinear average of a driven, dissipative, irreversible process reproduces an equilibrium quantity exactly, which is a much stranger and stronger claim.
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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