# Process function
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{{Short description|Thermodynamic quantity}}
{{about|the mathematical concept used in thermodynamics|the engineering indicator|process variable}}
{{Multiple issues|{{improve categories|date=November 2025}}{{single source|date=November 2025}}}}
{{Thermodynamics|cTopic=sysprop}}
{{see also|List of thermodynamic properties}}
In [thermodynamics](/source/thermodynamics), a [quantity](/source/quantity) that is well defined so as to describe the path of a process through the [equilibrium state](/source/equilibrium_state) space of a [thermodynamic system](/source/thermodynamic_system) is termed a '''process function''',[{{cite book |last=Sychev |first=V. V. |title=The Differential Equations of Thermodynamics |year=1991 |publisher=Taylor & Francis |isbn=978-1560321217}}] or, alternatively, a '''process quantity''', or a '''path function'''. As an example, [mechanical work](/source/mechanical_work) and [heat](/source/heat) are process functions because they describe quantitatively the transition between equilibrium states of a thermodynamic system.
==Path functions==
Path functions depend on the path taken to reach one state from another. Different routes give different quantities. Examples of path functions include [work](/source/work_(thermodynamics)), [heat](/source/heat) and [arc length](/source/arc_length). In contrast to path functions, [state function](/source/state_function)s are independent of the path taken. Thermodynamic [state variables](/source/State_function) are point functions, differing from path functions. For a given state, considered as a point, there is a definite value for each state variable and state function.
==Differentials==
Infinitesimal changes in a process function {{mvar|X}} are often indicated by {{mvar|δX}} to distinguish them from infinitesimal changes in a state function {{mvar|Y}} which is written {{mvar|dY}}. The quantity {{mvar|dY}} is an [exact differential](/source/exact_differential), while {{mvar|δX}} is not, it is an [inexact differential](/source/inexact_differential). Infinitesimal changes in a process function may be integrated, but the integral between two states depends on the particular path taken between the two states, whereas the integral of a state function is simply the difference of the state functions at the two points, independent of the path taken.
==Holonomic or non-holonomic==
In general, a process function {{mvar|X}} may be either [holonomic](/source/Holonomic_constraints) or non-holonomic. For a holonomic process function, an auxiliary state function (or integrating factor) {{mvar|λ}} may be defined such that {{math|''Y'' {{=}} ''λX''}} is a state function. For a non-holonomic process function, no such function may be defined. In other words, for a holonomic process function, {{mvar|λ}} may be defined such that {{math|''dY'' {{=}} ''λδX''}} is an exact differential. For example, thermodynamic work is a holonomic process function since the integrating factor {{math|''λ'' {{=}} {{sfrac|1|''p''}}}} (where {{mvar|p}} is pressure) will yield exact differential of the volume state function {{math|''dV'' {{=}} {{sfrac|''δW''|''p''}}}}. The [second law of thermodynamics](/source/second_law_of_thermodynamics) as stated by [Carathéodory](/source/Carath%C3%A9odory) essentially amounts to the statement that heat is a holonomic process function since the integrating factor {{math|''λ'' {{=}} {{sfrac|1|''T''}}}} (where {{mvar|T}} is temperature) will yield the exact differential of an entropy state function {{mvar|''dS'' {{=}} {{sfrac|''δQ''|''T''}}}}.