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I know that $δ$ sometimes represents the Dirac delta function but in my book it states "Suppose that equilibrium has been established Then a slight change in the position of the piston should not change the free energy since it is at a minimum that is $δA=0$" but in terms of this what exactly does it mean?

Qmechanic
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Astronomical
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2 Answers2

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I know that $\delta$ represents the Dirac delta function but in my book it states "Suppose that equilibrium has been established Then a slight change in the position of the piston should not change the free energy since it is at a minimum that is δA=0" but in terms of this what exactly does it mean.

Here, the symbol $\delta$ is being used to indicate "change." You can read $\delta A$ as the "change in Free Energy."

When a function like $A$ is at a maximum or a minimum (or saddle point) the function is stationary, meaning it doesn't change at first order when its function arguments change.

Since the Free Energy $A$ is a function of $T$, $V$, and $N$, we can take $$\delta A =0 $$ as effectively equivalent to: $$ \frac{\partial A}{\partial T} = 0\; $$ $$ \frac{\partial A}{\partial V} = 0\; $$ $$ \frac{\partial A}{\partial N} = 0\; $$

This is because, at first order: $$ \delta A = \frac{\partial A}{\partial N}\delta N + \frac{\partial A}{\partial V}\delta V + \frac{\partial A}{\partial T}\delta T\;. $$

hft
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That $\delta$ is not a Dirac delta, but the symbol for indicating a differential form that is an inexact differential, i.e. a differential whose integration depends on the path of integration,

$\displaystyle \int_{\ell^1_{A\rightarrow B}} \delta f \ne \displaystyle \int_{\ell^2_{A\rightarrow B}} \delta f \qquad$ in general, for different integration paths $\ell^i_{A\rightarrow B}$ with the same extreme points $A$, $B$

and not only on the values of a function (its primitive) at the extreme points of the integration path, like exact differentials

$\displaystyle \int_{\ell_{A\rightarrow B}} d f = f(B) - f(A) \qquad $ for every integration path with extreme points $A$ and $B$.

basics
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