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In `F33f`_`[thermodynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Thermodynamics]`_`f, a `F33f`_`[quantity`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Quantity]`_`f that is well defined so as to describe the path of a process through the `F33f`_`[equilibrium state`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Equilibrium_state]`_`f space of a `F33f`_`[thermodynamic system`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Thermodynamic_system]`_`f is termed a `!process function`!,`:cite-ref-sychev1991-1-0[`F5bf`_`[1`#cite-note-sychev1991-1]`_`f] or, alternatively, a `!process quantity`!, or a `!path function`!. As an example, `F33f`_`[mechanical work`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mechanical_work]`_`f and `F33f`_`[heat`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Heat]`_`f are process functions because they describe quantitatively the transition between equilibrium states of a thermodynamic system.

Path functions depend on the path taken to reach one state from another. Different routes give different quantities. Examples of path functions include `F33f`_`[work`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Work_(thermodynamics)]`_`f, `F33f`_`[heat`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Heat]`_`f and `F33f`_`[arc length`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Arc_length]`_`f. In contrast to path functions, `F33f`_`[state functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=State_function]`_`f are independent of the path taken. Thermodynamic `F33f`_`[state variables`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=State_function]`_`f 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.

Infinitesimal changes in a process function X are often indicated by δX to distinguish them from infinitesimal changes in a state function Y which is written dY. The quantity dY is an `F33f`_`[exact differential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exact_differential]`_`f, while δX is not, it is an `F33f`_`[inexact differential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inexact_differential]`_`f. 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.

In general, a process function X may be either `F33f`_`[holonomic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Holonomic_constraints]`_`f or non-holonomic. For a holonomic process function, an auxiliary state function (or integrating factor) λ may be defined such that `*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, λ may be defined such that `*dY`* = `*λδX`* is an exact differential. For example, thermodynamic work is a holonomic process function since the integrating factor `*λ`* = ⁠1/`*p`*⁠ (where p is pressure) will yield exact differential of the volume state function `*dV`* = ⁠`*δW`*/`*p`*⁠. The `F33f`_`[second law of thermodynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Second_law_of_thermodynamics]`_`f as stated by `F33f`_`[Carathéodory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Carathéodory]`_`f essentially amounts to the statement that heat is a holonomic process function since the integrating factor `*λ`* = ⁠1/`*T`*⁠ (where T is temperature) will yield the exact differential of an entropy state function `*dS`* = ⁠`*δQ`*/`*T`*⁠.`:cite-ref-sychev1991-1-1[`F5bf`_`[1`#cite-note-sychev1991-1]`_`f]

>>Contents

• `F0af`_`[References`#references]`_`f

• `F0af`_`[See also`#see-also]`_`f

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>>References

`:cite-note-sychev1991-1`!1.`! `F0af`_`[↑`#cite-ref-sychev1991-1-0]`_`f `:citerefsychev1991`aSychev, V. V. (1991). `*The Differential Equations of Thermodynamics`*. Taylor & Francis. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1560321217.

>>See also

• `F33f`_`[Thermodynamics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Thermodynamics]`_`f

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