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+---
+title: "Fundamental relation of thermodynamics"
+sort_title: "Fundamental relation of thermodynamics"
+date: 2024-07-21 # Originally 2021-07-07, major rewrite
+categories:
+- Physics
+- Thermodynamics
+layout: "concept"
+---
+
+In most areas of physics,
+we observe and analyze the behaviour
+of physical systems that have been "disturbed" some way,
+i.e. we try to understand what is *happening*.
+In thermodynamics, however,
+we start paying attention once the disturbance has ended,
+and the system has had some time to settle down:
+when nothing seems to be happening anymore.
+
+Then a common observation is that the system "forgets" what happened earlier,
+and settles into a so-called **equilibrium state**
+that appears to be independent of its history.
+No matter in what way you stir your tea, once you finish,
+eventually the liquid stops moving, cools down,
+and just... sits there, doing nothing.
+But how does it "choose" this equilibrium state?
+
+
+
+## Thermodynamic equilibrium
+
+This history-independence suggests that equilibrium
+is determined by only a few parameters of the system.
+Prime candidates are the **mole numbers** $$N_1, N_2, ..., N_n$$
+of each of the $$n$$ different types of particles in the system,
+and its **volume** $$V$$.
+Furthermore, the microscopic dynamics
+are driven by energy differences between components,
+and obey the universal principle of energy conservation,
+so it also sounds reasonable to define a total
+**internal energy** $$U$$.
+
+Thanks to many decades of empirical confirmations,
+we now know that the above arguments can be combined into a postulate:
+the equilibrium state of a closed system with fixed $$U$$, $$V$$ and $$N_i$$
+is completely determined by those parameters.
+The system then "finds" the equilibrium
+by varying its microscopic degrees of freedom
+such that the **entropy** $$S$$ is maximized
+subject to the given values of $$U$$, $$V$$ and $$N_i$$.
+This statement serves as a definition of $$S$$,
+and explains the **second law of thermodynamics**:
+the total entropy never decreases.
+
+We do not care about those microscopic degrees of freedom,
+but we do care about how $$U$$, $$V$$ and $$N_i$$ influence the equilibrium.
+For a given system, we want a formula $$S(U, V, N_1, ..., N_n)$$,
+which contains all thermodynamic information about the system
+and is therefore known as its **fundamental relation**.
+
+The next part of our definition of $$S$$
+is that it must be invertible with respect to $$U$$,
+meaning we can rearrange the fundamental relation
+to $$U(S, V, N_1, ... N_n)$$ without losing any information.
+Specifically, this means that $$S$$ must be continuous,
+differentiable, and monotonically increasing with $$U$$,
+such that $$S(U)$$ can be inverted to $$U(S)$$ and vice versa.
+
+The idea here is that maximizing $$S$$ at fixed $$U$$
+should be equivalent to minimizing $$U$$ for a given $$S$$
+(we prove this later).
+Often it is mathematically more convenient
+to choose one over the other,
+but by definition both approaches are equally valid.
+And because $$S$$ is rather abstract,
+it may be preferable to treat it as a parameter
+for a more intuitive quantity like $$U$$.
+
+Next, we demand that $$S$$ is additive over subsystems,
+so $$S = S_1 + S_2 + ...$$, with $$S_1$$ being the entropy of subsystem 1, etc.
+Consequently, $$S$$ is an **extensive** quantity of the system,
+just like $$U$$ (and $$V$$ and $$N_i$$),
+meaning they satisfy for any constant $$\lambda$$:
+
+$$\begin{aligned}
+    S(\lambda U, \lambda V, \lambda N_1, ..., \lambda N_n)
+    &= \lambda S(U, V, N_1, ..., N_n)
+    \\
+    U(\lambda S, \lambda V, \lambda N_1, ..., \lambda N_n)
+    &= \lambda U(S, V, N_1, ..., N_n)
+\end{aligned}$$
+
+For $$U$$, this makes intuitive sense:
+the total energy in two identical systems
+is double the energy of a single of those systems.
+Actually, reality is a bit hazier than this:
+dynamics are governed by energy *differences* only,
+so an offset $$U_0$$ can be added without a consequence.
+We should choose an offset and a way to split the system into subsystems
+such that the above relation holds for our convenience.
+Fortunately, this choice often makes itself.
+
+$$S$$ does not suffer from this ambiguity,
+since the **third law of thermodynamics** clearly defines
+where $$S = 0$$ should occur: at a temperature of absolute zero.
+In this article we will not explore the reason for this requirement,
+which is also known as the **Nernst postulate**.
+Furthermore, in most situations this law can simply be ignored.
+
+Since $$U$$, $$S$$, $$V$$ and $$N_i$$ are all extensive,
+the partial derivatives of the fundamental relation are **intensive** quantities,
+meaning they do not depend on the size of the system.
+Those derivatives are very important,
+since they are usually the equilibrium properties we want to find.
+
+
+
+## Energy representation
+
+When we have a fundamental relation of the form $$U(S, V, N_1, ..., N_n)$$,
+we say we are treating the system's thermodynamics
+in the **energy representation**.
+
+The following derivatives of $$U$$ are used as the thermodynamic *definitions*
+of the **temperature** $$T$$, the **pressure** $$P$$,
+and the **chemical potential** $$\mu_k$$ of the $$k$$th particle species:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            T
+            &\equiv \bigg( \pdv{U}{S} \bigg)_{V, N_i}
+            \\
+            P
+            &\equiv - \bigg( \pdv{U}{V} \bigg)_{S, N_i}
+            \\
+            \mu_k
+            &\equiv \bigg( \pdv{U}{N_k} \bigg)_{S, V, N_{i \neq k}}
+        \end{aligned}
+    }
+\end{aligned}$$
+
+The resulting expressions of the form $$T(S, V, N_1, ..., N_n)$$ etc.
+are known as the **equations of state** of the system.
+Unlike the fundamental relation, a single equation of state
+is not a complete thermodynamic description of the system.
+However, if *all* equations of state are known
+(for $$T$$, $$P$$, and all $$\mu_k$$),
+then the fundamental relation can be reconstructed.
+
+As explained above, physical dynamics are driven by energy differences only,
+so we expand an infinitesimal difference $$\dd{U}$$ as:
+
+$$\begin{aligned}
+    \dd{U}
+    = \bigg( \pdv{U}{S} \bigg)_{V, N_i} \!\dd{S}
+    \:\:+\:\: \bigg( \pdv{U}{V} \bigg)_{S, N_i} \!\dd{V}
+    \:\:+\:\: \sum_{k}^{} \bigg( \pdv{U}{N_k} \bigg)_{S, V, N_{i \neq k}} \!\dd{N_k}
+\end{aligned}$$
+
+Those partial derivatives look familiar.
+Substituting $$T$$, $$P$$ and $$\mu_k$$ gives a result
+that is also called the **fundamental relation of thermodynamics**
+(as opposed to the fundamental relation of the system only,
+just to make things confusing):
+
+$$\begin{aligned}
+    \boxed{
+        \dd{U}
+        = T \dd{S} - P \dd{V} + \sum_{k}^{} \mu_k \dd{N_k}
+    }
+\end{aligned}$$
+
+Where the first term represents heating/cooling (also written as $$\dd{Q}$$),
+and the second is physical work done on the system
+by compression/expansion (also written as $$\dd{W}$$).
+The third term is the energy change due to matter transfer and is often neglected.
+Hence this relation can be treated as a form
+of the **first law of thermodynamics** $$\Delta U = \Delta Q + \Delta W$$.
+
+Because $$T$$, $$P$$ and $$\mu_k$$ generally depend on $$S$$, $$V$$ and $$N_k$$,
+integrating the fundamental relation can be tricky.
+Fortunately, the fact that $$U$$ is extensive offers a shortcut.
+Recall that:
+
+$$\begin{aligned}
+    \lambda U(S, V, N_1, ..., N_n)
+    &= U(\lambda S, \lambda V, \lambda N_1, ..., \lambda N_n)
+\end{aligned}$$
+
+For any $$\lambda$$.
+Let us differentiate this equation with respect to $$\lambda$$, yielding:
+
+$$\begin{aligned}
+    U
+    &= \pdv{}{\lambda} U(\lambda S, \lambda V, \lambda N_1, ..., \lambda N_n)
+    \\
+    &= \pdv{U(\lambda S)}{(\lambda S)} \pdv{(\lambda S)}{\lambda}
+    + \pdv{U(\lambda V)}{(\lambda V)} \pdv{(\lambda V)}{\lambda}
+    + \sum_{k} \pdv{U(\lambda N_k)}{(\lambda N_k)} \pdv{(\lambda N_k)}{\lambda}
+    \\
+    &= \pdv{U(S)}{S} S
+    + \pdv{U(V)}{V} V
+    + \sum_{k} \pdv{U(N_k)}{N_k} N_k
+\end{aligned}$$
+
+Where we once again recognize the derivatives.
+The resulting equation is known as the **Euler form**
+of the fundamental relation of thermodynamics:
+
+$$\begin{aligned}
+    \boxed{
+        U
+        = T S - P V + \sum_{k} \mu_k N_k
+    }
+\end{aligned}$$
+
+Plus a constant $$U_0$$ of course,
+although $$U_0 = 0$$ is the most straightforward choice.
+
+
+
+## Entropy representation
+
+If the system's fundamental relation
+instead has the form $$S(U, V, N_1, ..., N_i)$$,
+we are treating it in the **entropy representation**.
+Isolating the above fundamental relation of thermodynamics
+for $$\dd{S}$$ yields its equivalent form in this representation:
+
+$$\begin{aligned}
+    \boxed{
+        \dd{S}
+        = \frac{1}{T} \dd{U} + \frac{P}{T} \dd{V} - \sum_{k}^{} \frac{\mu_k}{T} \dd{N_k}
+    }
+\end{aligned}$$
+
+From which we can then read off the standard partial derivatives of $$S(U, V, N_1, ..., N_n)$$:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            \frac{1}{T}
+            &= \bigg( \pdv{S}{U} \bigg)_{V, N_i}
+            \\
+            \frac{P}{T}
+            &= \bigg( \pdv{S}{V} \bigg)_{U, N_i}
+            \\
+            \frac{\mu_k}{T}
+            &= - \bigg( \pdv{S}{N_k} \bigg)_{U, V, N_{i \neq k}}
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Note the signs: the parameters $$U$$, $$V$$ and $$N_i$$ are implicitly related
+by our requirement that $$S$$ is stationary at a maximum,
+so the [triple product rule](/know/concept/triple-product-rule/)
+must be used, which brings some perhaps surprising sign changes.
+Reading them off in this way is easier.
+
+And of course, since $$S$$ is defined to be an extensive quantity,
+it also has an Euler form:
+
+$$\begin{aligned}
+    \boxed{
+        S
+        = \frac{1}{T} U + \frac{P}{T} V - \sum_{k} \frac{\mu_k}{T} N_k
+    }
+\end{aligned}$$
+
+Finally, it is worth proving that minimizing $$U$$
+is indeed equivalent to maximizing $$S$$.
+For simplicity, we consider a system
+where only the volume $$V$$ can change
+in order to reach an equilibrium;
+the proof is analogous for all other parameters.
+Clearly, $$S$$ is stationary at its maximum:
+
+$$\begin{aligned}
+    0
+    &= \bigg( \pdv{S}{V} \bigg)_{U, N_i}
+    = - \frac{ \bigg( \displaystyle\pdv{U}{V} \bigg)_{S, N_i} }{ \bigg( \displaystyle\pdv{U}{S} \bigg)_{V, N_i} }
+    = - \frac{1}{T} \bigg( \pdv{U}{V} \bigg)_{S, N_i}
+\end{aligned}$$
+
+Where we have used the triple product rule.
+This can only hold if $$(\ipdv{U}{S})_{S, N_i} = 0$$,
+meaning $$U$$ is also at an extremum.
+But $$S$$ is not just at any extremum: it is at a *maximum*, so:
+
+$$\begin{aligned}
+    0
+    > \bigg( \pdvn{2}{S}{V} \bigg)_{U, N_i}
+    &= \bigg( \pdv{}{V} \Big( \frac{P}{T} \Big) \bigg)_{U, N_i}
+    \\
+    &= \bigg( \pdv{}{V} \Big( \frac{P}{T} \Big) \bigg)_{S, N_i}
+    + \bigg( \pdv{}{S} \Big( \frac{P}{T} \Big) \bigg)_{V, N_i} \bigg( \pdv{S}{V} \bigg)_{U, N_i}
+    \\
+    &= \bigg( \pdv{}{V} \Big( \frac{P}{T} \Big) \bigg)_{S, N_i}
+    + \frac{P}{T} \bigg( \pdv{}{S} \Big( \frac{P}{T} \Big) \bigg)_{V, N_i}
+    \\
+    &= \frac{1}{T} \bigg( \pdv{P}{V} \bigg)_{S, N_i}
+    - \frac{P}{T^2} \bigg( \pdv{T}{V} \bigg)_{S, N_i}
+    + \frac{P}{T} \bigg( \pdv{}{S} \Big( \frac{P}{T} \Big) \bigg)_{V, N_i}
+    \\
+    &= - \frac{1}{T} \bigg( \pdvn{2}{U}{V} \bigg)_{S, N_i}
+    + \frac{P}{T} \bigg[ \bigg( \pdv{}{S} \Big( \frac{P}{T} \Big) \bigg)_{V, N_i}
+    - \frac{1}{T} \bigg( \pdv{T}{V} \bigg)_{S, N_i} \bigg]
+\end{aligned}$$
+
+Because $$S$$ is at a maximum, we know that $$P/T = 0$$,
+and $$T$$ is always above absolute zero
+(since we defined $$S$$ to be monotonically increasing with $$U$$),
+which leaves $$(\ipdvn{2}{U}{V})_{S, N_i} > 0$$
+as the only way to satisfy this inequality.
+In other words, $$U$$ is at a minimum, as expected.
+
+
+
+## References
+1.  H.B. Callen,
+    *Thermodynamics and an introduction to thermostatistics*, 2nd edition,
+    Wiley.
+2.  H. Gould, J. Tobochnik,
+    *Statistical and thermal physics*, 2nd edition,
+    Princeton.