I've come across the Onsager reciprocal principle. It's almost clear, except for thermodynamic conjugate quantities - what's that, physical meaning (except the formal definitions: $X_i = -\frac{1}{k}\frac{\partial S}{\partial x_i}$, which isn't clear) and why:
\begin{equation} \langle X_i\cdot x_k\rangle = \delta_{ik} \end{equation}
The Wikipedia is lack for references in this article.
The fundamental quantity in thermodynamics is entropy, which is a function of $n$-variables $S=S(x_1, x_2,...,x_n)$. For instance, for a simple mono-component system $S=S(U,V,N)$ where $U$ is internal energy, $V$ is volume, and $N$ composition.
Taking the differential
$$\mathrm{d}S = \sum_i \left( \frac{\partial S}{\partial x_i}\right)_{j \neq i} \mathrm{d}x_i = \sum_i F_i \mathrm{d}x_i$$
The quantities $F_i \equiv ({\partial S}/{\partial x_i})_{j \neq i} $ are intensive entropic parameters and measure the change in entropy when variables change. For instance the intensive entropic parameter $(1/T)$ gives the change on entropy due to a change in the energy $U$.
Using the thermodynamic theory of fluctuations it can be shown that
$$F_i = k \left( \frac{\partial \ln P}{\partial x_i}\right)$$
where $P$ is the probability of a fluctuation in the variables near an equilibrium state.
Using the definition of average
$$\langle A \rangle = \int A P \mathrm{d}x_1 \mathrm{d}x_2 \cdots \mathrm{d}x_n$$
the demonstration of the central result of linear nonequilibrium thermodynamics
$$\langle F_i \cdot x_j \rangle = -k \delta_{ij} $$
is direct although it needs first the use of $(\partial \ln P / \partial x_i)P = \partial P / \partial x_i$ in the integrand and next integration by parts.
What Wikipedia makes is to rewrite this central result using the new quantities $X_i \equiv - F_i / k$ but the physically important quantities are the $F_i$ often also called in this context thermodynamic forces or affinities.
Thermodynamic conjugate variables are pairs of variables $x_i$, $X_i$ where the product $X_i\cdot dx_i$ has the dimension of energy and actually appears as a term in the infinitesimal variation of energy, free energy, or work such as $dE$.
In the pair, one quantity is intensive and the other is extensive.
The best example is pressure and volume because $p\,dV$ is a term in $dE$. Analogously, temperature and entropy because of $T\,dS$, chemical potential and particle number because of $\mu \,dN$, and many electric, magnetic, gravitational, and surface tension examples exist.
If you have $$ dE = \dots + p\,dV, $$ it's easy to see that $p$ is the partial derivative of $E$ with respect to $V$. This role of $p,V$ may be reverted if you consider $H=E+pV$ instead of $E$ i.e. substitute $E = H-pV$. Then you will get $$dH = \dots - V\,dp.$$ This "Lagrange duality" may be applied to any pair.
