In the one-dimensional growth problem, a crucial quantity is represented by \(W^*(\sigma)-\varrho M_B\), where \(W^*(\sigma)\) is the Legendre transform of the free energy \(W(\lambda)\) per unit referential length and \(\varrho M_B\) is the chemical energy needed to create a length of solid material.
The equation governing accretion tells us that when a reference unit length of solid material is created, the amount of energy provided from the exterior is equal to \(W^*(\sigma)-M_B\varrho\).
The dissipation rate turns out to be \[\delta=(W^*(\sigma)-M\varrho+\mu\varrho)V,\] where \(V\) is the accretion velocity, \(\mu\) is the chemical potential of the free monomers, \(M\) is the chemical energy of the bound monomers, and \(W^*(\sigma)\) is the Legendre transform of the strain energy \(W(\sigma)\).
In thermodynamics, the equivalent of the strain energy \(W(\sigma)\) is the free energy \[F(T,V,N)=U(S,V,N)-TS.\] In thermodynamics the Legendre transform of the free energy with respect to \(V\) is the Gibbs free energy, defined as \[G(T,p,N)=F(T,V,N)-pV.\] Thus, if the quantities \(\sigma\) and \(\lambda\) correspond, respectively, to pressure and volume, we may argue that the thermodynamical equivalent of the Legendre transform \(W^*(\sigma)\) is the Gibbs free energy. Thus, it would seem that the driving force for accretion of a thermodynamic system with a variable number of elements is \[G-\mu\Omega,\] where \(\Omega\) is the specific volume of a particle in the reference state.