In mathematics the concept of 'a group' is used to formulate, in a most general way, any kind of mathematical operation, for example, addition, multiplication or rotation. Generally, a mathematical group consists of 'elements' *a*, *b*, *c*, ... and an instruction that attributes to each pair of elements, say *a* and *b*, a new element a\circ b in such a way that three group axioms G1, G2 and G3 strictly apply. Abelian groups are commutative for which group axiom G4 applies as well:

\begin{array}{cc}\text{Foranythreeelements}a,b\text{and}c\text{associativityapplies}:& \left(a\circ b\right)\circ c=a\circ \left(b\circ c\right)\end{array}

(G1)

\begin{array}{cc}\text{Foranyelement}a\text{thereisanidentityelement}e:& e\circ a=a\circ e=a\end{array}

(G2)

\begin{array}{cc}\text{Foranyelement}a\text{thereisaninverseelement}{a}^{\u2013\text{1}}:& a\circ {a}^{\u2013\text{1}}={a}^{\u2013\text{1}}\circ a=e\end{array}

(G3)

\begin{array}{cc}\text{Anypairofelements}a\text{and}b\text{arecommutative}:& a\circ b=b\circ a\end{array}

(G4)

The mathematical operation of addition in conjunction with elements that are integers, {*a*, *b*, *c*, ...} ∈ ℤ, is an abelian group. To specify, replace the above general formalism with a specific one, i.e., replace \circ with +, *e* with 0, and *a*^{–1} with –*a*. The mathematical operation of multiplication in conjunction with elements that are positive real numbers, {*a*, *b*, *c*, ...} ∈ ℝ^{+}, is also an abelian group: \circ :=\cdot ,e:=1, and {a}^{-1}:=1/a. Note that, whereas the addition of real numbers is an abelian group, axiom G3 usually does not apply to the multiplication of positive integers {*a*, *b*, *c*, ...} ∈ ℤ^{+} because generally {*a*^{–1}, *b*^{–1}, *c*^{–1}, ...} ∉ ℤ. Most commonly, mathematical groups are used to describe geometrical symmetries; the vast majority of mathematical groups are non-abelian (axiom G4 does not apply), in particular, when the parameter space is higher than 2D (two-dimensional).

In physics and physical cosmology this formalism is extremely useful for the description of spontaneous symmetry breaking into different fundamental forces through the application of gauge theories, such as in Quantum Electrodynamics (QED) or Quantum Chromodynamics (QCD) being the basis of the Standard Model. A gauge theory is a type of field theory in which the Lagrangian is invariant under a certain continuous group of local transformations. The Lagrangian of a dynamical system is a function that summarises the dynamics of the system. Common to all these theories is the utilization of Legendre transformations. The Legendre transformation can be generalized to the Legendre-Fenchel transformation. It is commonly used in thermodynamics and in the Hamiltonian formulation of classical mechanics, for example, to describe various thermodynamic potentials in classical thermodynamics, to derive partition functions from a state equation in statistical thermodynamics, to link Lagrange (classical) mechanics with Hamiltonian (quantum) mechanics, to formulate QED and QCD, and more. In a thermodynamic context the linking of energy with entropy through a quantified similarity argument, as described in group thermodynamics [1], may be viewed as a Legendre transformation on a group, rather than individual, level.

Consider a mathematically general relationship between difference and ratio (quotient) being the operational results obtained from, respectively, the substraction and division of the same elements, say, *x* and *y* (eqns. 1 and 2):

Interpret *x* and *y* as any two parameters that are needed to define the characteristics of some state and *v* as a mathematically independent variable. For example, {*x*, *y*} may be the coordinates of a point in 2D space and *v* a force acting on one coordinate (in one direction) but not the other. If *x* were to represent the distance between two objects in Euclidian 3D space, difference *u* would describe the movement of these objects through spacetime according to Einstein's special relativity \text{d}{s}^{\text{2}}=\text{d}{x}_{\text{i}}^{2}-{\text{c}}^{\text{2}}\cdot \text{d}{t}^{\text{2}} where d = differential, *s* = distance in spacetime, *x*_{i} = spatial Cartesian coordinates {*x*, *y*, *z*}, c = vacuum speed of light, and *t* = time. Difference *u* could also represent the incremental change Δ in free energy upon transformation of one (metastable) macrostate into another, as expressed in the Gibbs-Helmholtz equation \Delta {G}_{\text{T}}=\Delta H-T\cdot \Delta S, where \Delta {G}_{\text{T}} = Gibbs free energy change at temperature *T*, Δ*H* = energy change at constant pressure (enthalpy change), and Δ*S* = entropy change. Equation 2 describes in a generalised fashion the relationship between difference *u* and ratio *x*/*y* as depicted in Figure 1. Note that *v* is the root of the function in equation 2.

To analyse more readily, a set of 2D projections of the same function is depicted in Figure 2 where *u* is sampled over a representative range of numerical values for *x* and *y*. Note that, trivially, *x*/*y* requires for *v* > 0 the root to lie in the quadrants where both *x* and *y* have the same sign (right half of the 2D plot), whereas a negative value for *v* places the root in the quadrants of opposite signs for *x* and *y* (reflection of the hyperbles through *x*/*y* = 0, not shown).

In special relativity the relationship between difference and ratio immediately visualises how the shape of squared distance differentials in spacetime d*s*^{2} (ordinate in the above 2D plot) relates to the ratio of squared spatial distance differentials over squared time differentials \text{d}{x}_{\text{i}}^{2}\text{/d}{t}^{\text{2}} for *v* = c^{2} (abscissa). In thermodynamics the above relationship shows how changes in Gibbs free energy Δ*G*_{T} relate to the temperature at which Δ*G* vanishes, *T* =Δ*H*/Δ*S* for Δ*C*_{p} = 0, or in a differential form for infinitesimal state changes: d*G*_{T} vs. *T* = d*H*/d*S*. It was shown that the Δ*H* and Δ*S* values of similar objects at their respective equilibrium temperatures are all linearly related to their corresponding Δ*G*_{T} values [1]. These realised experimental values of similar objects gather linearly, in the above plot, close to the root of the function at {*x* – *v*·*y* = 0, *x*/*y* = *v*}, i.e., close to the saddle point. Physics imposes positive *v* values in both special-relativity and thermodynamics, since there is no immediate physical meaning for a negative speed of light (reaching *in vacuo* a maximum constant positive value c) nor negative absolute temperature, both, in accord with the Second Law.

General and special relativity describe the shape of and the dynamics within a spacetime where the objects may or may not move in a reversible fashion. The main focus in relativity theory is not reversibility but rather to describe objects (particles) and their 'directed' movements at up to relativistic speeds along geodesics in spacetime being shaped by gravitational fields. Thermodynamics describes the energetics of dynamic systems that contain multi-component objects ('macrostates') able to adopt, essentially reversibly, multiple isoergonic (degenerate) 'microstates'. Both theories are, in terms of physics, quite different from one another. The former is based on Einstein's mass-energy equivalence; in the latter an energy-entropy equivalence of macrostates was identified within groups of similar objects. Mathematically, the former is generated from applying Pythagorean rules to the coordinates of a 4D Lorentzian geometry (two sums and a difference between four positive terms: \text{d}{x}_{1}^{2}+\text{d}{x}_{2}^{2}+\text{d}{x}_{3}^{2}-{\text{c}}^{\text{2}}\cdot \text{d}{t}^{\text{2}}). The latter originates from correlating and partitioning energy contributions through Legendre transformations. In spite of this apparent disparity, the mathematical formalism looks alike for both theories. Lorentzian geometry and the Gibbs-Helmholtz equation are both typified through a crucial substraction term; through the *mathematical difference viz. the physical balance* between two most fundamental components of spacetime and, respectively, free energy. What is the squared spatial distance differential in special relativity, d*x*_{i}^{2}, is formally the energy differential in thermodynamics, d*U* (or d*H* at constant pressure). The spacetime differential that describes changes in time not space, c^{2}·d*t*^{2}, formally becomes the entropic component of the free energy differential, *T*·d*S*. The balance between each pair of components gives rise to the squared spatio-temporal distance differential d*s*^{2} and, respectively, the free energy differential d*F* (or d*G* at constant pressure).