VARIOUS VARIABLES
In electric-network analogies, variables are classified either as “longitudinal” or “transversal”. Longitudinal
variables are those that play the role of currents in the networks; e.g., currents, mass flows, volume flows,
thermal powers, forces in force-current analogies, and velocities in force-voltage analogies. Transversal
variables are those that play the role of voltages; e.g., voltages, pressures, temperatures, velocities in forcecurrent analogies, and forces in force-voltage analogies. An interesting issue is the possibility to produce
electric-network analogies of electric networks by swapping variables; i.e., to produce “dual networks” with
voltages as longitudinal variables and currents as transversal ones. This explains the duality of the forcecurrent and force-voltage analogies, and opens up various alternatives, such as having pressures as
longitudinal variables and mass flows as transversal ones, and so on.
Summarizing: some variables are classified as longitudinal and some others as transversal to produce
electric-network analogies; longitudinal and transversal variables can be swapped.
Thermodynamic texts classify variables in various ways. In one of them, variables are classified either as
“extensive” or “intensive”. Extensive variables are those “which are proportional to the size of the system”.
Such a deliciously flimsy definition gathers, e.g., volumes, masses, energies, enthalpies, entropies,
momentums and heat capacities. Intensive variables are those “that specify a local property”. Such an equally
flimsy definition captures, e.g., temperatures, pressures, specific volumes, specific energies, specific
enthalpies, specific entropies, specific momentums, specific heats, and mass densities, energy densities,
enthalpy densities, momentum densities and entropy densities. Yet, it is hard to classify mass flows, thermal
powers, mechanical powers and forces if one rigidly assumes that “size” means “volume”. Heats and works
are hard to classify in any case.
Summarizing: some variables are classified as extensive and some others as intensive in thermodynamic
texts; and some variables, though interesting, are a bit puzzling.
In another thermodynamic way, some variables are classified as “state variables”. State variables are those
that specify the states of systems. Such a comical definition seizes pressures, volumes and temperatures, to
start producing state equations (where either pressures or volumes or temperatures are considered state
functions while either volumes and temperatures, or temperatures and pressures, or pressures and volumes,
are considered true state variables); then it grabs masses and numbers of moles (with either masses or
numbers of moles as state functions and either numbers of moles or masses as true state variables, at least
provisionally); then it snatches internal energies and enthalpies (as state functions provisionally, but as true
state variables in other occasions, displacing some of those that were granted the honor before); then, it grips
entropies (in a similar fashion); and stresses; and chemical compositions and extents of reaction; and
concentrations; and states of aggregation; and velocities and kinetic energies; and a lot more. A clue to solve
the mystery of the ever-emerging state variables is that -Heaven knows why- the sets of mass-points are called
“systems” in Mechanics, and the sets of their positions and velocities at given instants of time are called their
“(dynamical) states”. Statistical Mechanics suggests that many of the thermodynamic variables reflect
statistical summaries (e.g., averages) of the sets of positions and velocities for wondrous sets of mass-points,
at given instants of time. Those are supposed to be the thermodynamic state variables and their growing set is
the thermodynamic state. The trouble is, as Enrico Fermi put it in 1936, “that the knowledge of the
thermodynamical state alone is by no means sufficient for the determination of the dynamical state… (e.g., in)
the thermodynamical state of a homogeneous fluid of given volume at a given temperature… there is an
infinite number of states of molecular motion that correspond to it… (and) the system exists successively in
all these dynamical states… From this point of view we may say that a thermodynamical state is the ensemble
of all the dynamical states through which… the system is rapidly passing… This definition of state is rather
abstract and not quite unique…”. Pretending more down-to-Earth definitions, a mysterious property of state
variables is often invoked (not in the thermodynamic sense of “property”, anyway): their integrals along all
thermodynamic paths equal 0. But, comically again, “thermodynamic paths”, also called trajectories or
transformations as if that abundance were better leading, are continuous successions of intermediate… states.
On the other hand, state variables are sometimes imagined associated with exact (“perfect”) differentials; i.e.,
with mathematical differential expressions of several “independent variables” which happen to mirror
mathematical continuous functions of those independent variables. This hints another mysterious property,
which indeed supersedes the prior one both in contents and comicality: thermodynamic state variables are
mathematical functions of… other thermodynamic state variables. Mathematical scalar fields resound here.
But a little contradiction appears in the horizon: volumes are not functions of temperatures and pressures
when masses and compositions are given; there is a problem, e.g., in condensation: a substance with constant
mass and composition, initially gaseous, becomes liquid at constant temperature if its pressure is constant, and
yet its volume varies. The consideration that volumes are state functions while temperatures and pressures are
true state variables, is dubious. Of course, that clouds the present paragraph from the start, just a little after
some variables were classified as “state variables”. And this reminds that, if some thermodynamic variables
are classified as state variables, some others might be classified differently. The problem is that, at least
classically, Thermodynamic does not seem to care about non-state variables. So, one would do well
questioning the reasons for the insistence on the classification and on words such as “states” and “systems”.
Summarizing: some variables are classified as state variables in thermodynamic texts; but the meaning of
the classification is obscure.
In state-space formulations, variables are classified either as “input variables” or “state variables” or “output
variables”. Input variables are those which are not state variables and appear as arguments in non-dynamical
operations within a set of equations (e.g., in additions, subtractions, multiplications,…). This definition
includes an interesting issue: non-dynamical operations must appear in the right-hand sides of equations. It
admits a huge constellation of variables, comprising currents, mass flows, volume flows, thermal powers,
mechanical powers, forces, velocities, voltages, pressures, temperatures, volumes, masses, energies,
enthalpies, entropies, momentums, heat capacities, specific volumes, specific energies, specific enthalpies,
specific entropies, specific momentums, specific heats, mass densities, energy densities, enthalpy densities,
momentum densities, entropy densities, heats, works, distances, numbers of moles, stresses, strains, chemical
compositions, extents of reaction, concentrations, states of aggregation, prices, supplied flows, demanded
flow, populations and so on and so forth. State variables are those appearing as arguments of time derivatives
(i.e., of rates of change). Such a definition includes another interesting issue: time derivatives must appear
alone in the left-hand sides of some equations, which are comically baptized as state equations. State variables
can also appear in the right-hand sides of equations (no matter whether these are state equations or not), but as
arguments of non-dynamical operations (like input variables). The definition accepts the same constellation. It
is hard to see how these state variables are related to their homonyms in the prior paragraph. Output variables
are those appearing alone in the left-hand sides of some equations comically known as output equations. This
definition tolerates the same constellation again, and suggests still another interesting issue: that no other
variables are allowed beyond the output variables, the state variables and the input variables.
Summarizing: variables are classified as either input variables or state variables or output variables in statespace formulations whose relation with thermodynamic states is perplexing; the absence of other classes of
variables is also bewildering.
The expressions “input variables”, “state variables” and “output variables” in state-space formulations name
the roles of the variables in a set of equations. They welcome any variable according to that circumstance.
The expressions “longitudinal variables” and “transversal variables” in electric-network analogies exhibit a
resembling behavior in a network diagram. The causal dream promotes another role-naming classification:
according to it, variables are classified either as “dominant” or “dependent” (or, equivalently in a substantive
manner, as “causes” or “effects”, “stimuli” or “responses”, or so on). However, the roles are supposed to be
played in reality rather than in its representations. The classification is definitely elucidated, not by observing
diagrams or equations, but by enforcing alternatives on the hypothetically dominant variables and then
observing the supposedly dependent ones; if these vary as thought, one decides that they are dependent and
the hypothetically dominant variables are truly so; otherwise, one decides differently. Anyway, the roles are
local and the same variable can be dominant on one side and dependent on another. Hopefully, the dependent
role should be reflected by a representation in the left-hand side of an equation while the dominant role should
be represented on the right-hand side. Thus, a variable can appear both as dominant and as dependent in a set
of equations. State variables in state-space formulations might be good candidates to illustrate this
ambivalence.
Summarizing: variables are classified as either dominant or dependent when conjecturing causality; the
classification is not trivial; ideally, it should be experimental and appear somehow in the representations.
Imagine a warship firing a battery of cannons one after the other. Each shell carries some mass and
momentum to damage the foes. Mass and momentum flows leave the ship, ruled by the artillery squad.
Imagine that the firing ceases and the last shell goes away through the air pursuing its target. Keep your eyes
on that shell and shrink your vision to look at half of it. Mass and momentum flow out of your sight as you
shrink your visual region. Mass and momentum are supposed to be eternals. They endure in the Universe.
They only vary inside regions. They do that because they flow into and/or out of some regions, and out of
and/or into some others. And they flow because the regions change and/or other relevant causes exist.
Thermodynamic texts seem close to this notion when classifying variables either as extensive or intensive.
Masses, for instance, are explicitly considered extensive in them; energies, too; and energy is also supposed to
be an eternal. But their definition of extensive variables as those “which are proportional to the size of the
system” does not fit: masses are proportional to the volumes of the regions only if mass density is
homogeneous. Moreover, mass flows are proportional to the areas of the boundaries if mass flow “density” is
homogeneous, but mass flow is not supposed to be eternal. On the other hand, electric-network analogies
seem close to the notion when classifying variables either as longitudinal or transversal. Mass flows, for
example, are explicitly considered longitudinal (in the pressure-voltage analogy); thermal powers, too (in
temperature-voltage analogy). But duality is devastating. By the same token, state-space formulations seem
close to the notion when classifying variables as either input variables or state variables or output variables.
Mass flows, e.g., dominate masses in such a way that the time derivatives (rates of change) of masses on the
left-hand sides of equations equal the (net) flows on the right-hand sides; so, masses appear in sets of
equations as state variables, and mass flows as input variables. But these depend on various causes (even on
such ethereal beings as the observer’s vision) in such a way that mass flows on the left-hand sides of
equations equal other variables; so, mass flows also appear in the sets of equations as output variables,
contradicting the state-space formulation. Usually, one can get rid of the contradiction by performing some
algebraic maneuvers that erase the masses (in general, the eternals) and/or the mass flows (in general, the
streams of eternals). The nuisance is that Algebra disregards causality and the erasures are heavy loses: all
kinds of derivatives (e.g., the rate of change of a pressure) emerge substituting the flows, the search of mere
correlations (e.g., thermodynamic state equations) jams the pursuit of dependences, and the whole scenario
becomes rather daunting.
Summarizing: eternals endure in the Universe and only vary inside regions because they flow from some of
them to others; the corresponding variables in the regions depend on the corresponding flows across the
boundaries; that gives us a start to build the causal dream on; for more, we can follow the flows.