State Variable ○꠹｜Definition｜1st｜20260912123148-00-⌔

State variable - Wikipedia

State variable

A state variable is one of the set of mathematical variables that are used to describe the “state” of a system, e.g. of a dynamical system or a thermodynamical system.

Intuitively, the state of a system is the minimum amount of information about the system which is required to describe the present properties of the system (at least from a certain point of view within a modelling decision of defining a system and separating the system from its environment; if a family of systems is modelled, where the systems are distinguished by parameters, then these parameters are usually not considered to be state variables for the systems, but external system parameters. E.g. physical masses or physical coupling constants for describing interaction strength are given external system parameters, but not state variables for the system.)¹

In the mathematical model, bounded observables are often taken to be elements of an algebra of observables.² Namely, if bounded observables, whose values might be added and multiplied, are modelled by using self-adjoint elements from a Cstar_algebra, then a state is a normalized positive element of the algebra’s dual (a continuous linear functional ρ): If the system is in a state $\rho$, then an observable $A$ provides an observed value $\rho (A)=<\rho ,A>$. (The brackets $<.,.>$ denote the duality pairing between the dual space of the algebra and the algebra itself.) For example, if the observable is a function $f$ from state space to real numbers, then evaluating the function at a point $x$ provides a state ${\rho }_x$ with $<{\rho }_x,f>=f(x)$, which is the point evaluation functional at point x. In this description, bounded state functions are certain bounded observables. It is possible to extend that notion to unbounded observables (examples in physics: a position component for full space, or a momentum component)³ for the general abstract case and therefore for more general unbounded state functions. Often, a maximal set of compatible commuting and independent observables can be chosen, and other compatible observables expressed as functions of these observables; then one has made a choice to express all compatible observables as functions of the independent observables, which become independent state variables.

For the definition and examples of states of a thermodynamic system see Thermodynamic state.

In thermodynamics, state variables are physically defined as large-scale characteristics or aggregate properties of a system which provide a macroscopic description of its states.⁴⁵ In general, state variables have the following properties in common:

• They don’t involve any special assumptions concerning the structure of matter, fields or radiation.
• They are few in number needed to describe the system.
• They are fundamental, as suggested by our sensory perceptions.
• They can be, in general, directly measured.⁴

Classical algebras of observables are Abelian (commutative), whereas algebras of quantum observables are Non-Abelian.⁶

The macroscopic thermodynamic description of microscopic quantum systems is obtained as a Thermodynamic Limit of the microscopic system, and a corresponding algebra of observables is obtained, which allows for a definition of macroscopic observables. A distinction between abstract macroscopic observables and observables as thermodynamic state functions can be obtained as follows: For a specific macroscopic state, one uses the GNS-representation to obtain a representation of the algebra of observables as linear operators on a Hilbert space, one chooses a maximally Abelian algebra of compatible observables and applies the Gelfand transform for Abelian C﹡-algebras to represent abstract observables as functions on a representation space of states (Gelfand space). In general, such a description as functions on a Gelfand space depends on the macroscopic state chosen and is therefore representation dependent. State equations establish certain relations between these functions and therefore might define a variety of states as a set defined by equations for variables. Often, this variety is finite dimensional and then can be described by choosing certain observable state functions as independent state variables and express the other state functions as functions of these independent variables. In this sense, the independent state variables are used to parametrize the variety of states as functions from their domain to the variety of states, whereas general state functions are functions from the set of states to real (or complex) numbers. Composing these functions expresses the values of general observables as functions of the independent state variables, which makes them dependent state variables.

For instance, by describing a thermodynamic system by using the canonical ensemble of microscopic systems in the thermodynamic limit provides independent state variables $T,V,N$ (temperature, volume, particle number) corresponding to the formula $dF=-SdT-pdV-\mu dN$ for the thermodynamic free energy $F=E-TS$ with S = entropy, E = internal energy, μ = chemical potential. All other state variables can then be expressed as functions of T,V,N: For a monatomic ideal gas, the state equation for the pressure is the ideal gas law and pressure becomes a dependent state variable, the equation for entropy is the Sackur-Tetrode equation, internal energy can be obtained by requiring Joule’s second law, and the chemical potential is computed via the Gibbs potential as described in the section Thermodynamic potentials of the article ideal gas.

All these variables depend only on the states, not on the history of how that state was obtained. This is an important property which makes them state variables, to be distinguished from e.g. process variables (better: process quantities), for which one cannot speak of attaining a particular value when the system is in a state. See the Non-Examples section below.

In the theory of Dynamical Systems, dynamical models that consist of coupled first-order differential equations are said to be in state-variable form.⁷

Printed 2026-09-12.

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Footnotes

1. Ludwig, Günther (1979). Einführung in die Grundlagen der Theoretischen Physik: Band 4: Makrosysteme, Physik und Mensch. Wiesbaden s.l: Vieweg+Teubner Verlag Imprint: Vieweg+Teubner Verlag. ISBN 978-3-663-12070-4. ↩

2. Emch, Gérard G. (2009). Algebraic methods in statistical mechanics and quantum field theory. Mineola, NY: Dover Publ. ISBN 978-0-486-47209-6. ↩

3. “Operators and Representation Theory”. Dover Publications. Retrieved 2026-06-07. ↩

4. Zemansky, Mark Waldo; Dittman, Richard (1997). Heat and thermodynamics: an intermediate textbook (7th ed.). New York: McGraw-Hill. ISBN 978-0-07-017059-9. ↩ ↩²

5. Sewell, Geoffrey L. (2014). Quantum theory of collective phenomena (Dover ed.). Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-78044-3. ↩

6. Thirring, Walter E. (2002). Quantum mathematical physics: atoms, molecules and large systems (2nd ed.). Berlin; New York: Springer. ISBN 978-3-540-43078-0. ↩

7. Palm, III William J. (2009). System Dynamics (2nd ed.). McGraw-Hill Medical Publishing. p. 420. ISBN 978-0-07-126779-3. ↩