In Norse mythology, Valhalla (/vælˈhælə,vɑːlˈhɑːlə/; from Old Norse: Valhǫll "hall of the slain") is a majestic, enormous hall located in Asgard, ruled over by the god Odin. Half of those who die in combat travel to Valhalla upon death, led by valkyries to be with Odin, while the other half gets chosen by the goddess Freyja for the field Fólkvangr. In Valhalla, the dead warriors join the masses of those killed in combat (known as the Einherjar) and various legendary Germanic heroes and kings, as they prepare to aid Odin during the events of Ragnarök. Before the hall stands the golden tree Glasir, and the hall's ceiling is thatched with golden shields. Various creatures live around Valhalla, such as the stag Eikþyrnir and the goat Heiðrún, described as standing atop Valhalla and consuming the foliage of the tree Læraðr.
Valhalla is attested in the Poetic Edda, compiled in the 13th century from earlier traditional sources, in the Prose Edda (written in the 13th century by Snorri Sturluson), in Heimskringla
(also written in the 13th century by Snorri Sturluson), and in stanzas
of an anonymous 10th century poem commemorating the death of Eric Bloodaxe known as Eiríksmál as compiled in Fagrskinna. Valhalla inspired innumerable works of art, publication titles, and elements of popular culture,
and is synonymous with a martial (or otherwise) hall of the chosen
dead. The name is rendered in modern Scandinavian languages as Valhöll in Icelandic, while the Swedish and Norwegian form is Valhall, in Faroese it's Valhøll, and in Danish it's Valhal.
Etymology
The Modern English noun Valhalla derives from Old Norse Valhǫll, a compound noun composed of two elements: the masculine noun valr 'the slain' and the feminine noun hǫll 'hall'. The form "Valhalla" comes from an attempt to clarify the grammatical gender of the word. Valr has cognates in other Germanic languages such as Old English wæl 'the slain, slaughter, carnage', Old Saxonwal-dād 'murder', Old High German 'battlefield, blood bath'. All of these forms descend from the Proto-Germanic masculine noun *walaz. Among related Old Norse concepts, valr also appears as the first element of the noun valkyrja 'chooser of the slain, valkyrie'.
The second element, hǫll, is a common Old Norse noun. It is cognate to Modern English hall and offers the same meaning. Both developed from Proto-Germanic *xallō or *hallō, meaning 'covered place, hall', from the Proto-Indo-European root *kol-. As philologists such as Calvert Watkins note, the same Indo-European root produced Old Norse hel, a proper noun employed for both the name of another afterlife location and a supernatural female entity as its overseer, as well as the modern English noun hell. In Swedish folklore, some mountains traditionally regarded as abodes of the dead were also called Valhall. According to many researchers, the hǫll element derives from hallr, "rock", and referred to an underworld, not a hall.
Attestations
Three valkyries bearing ale in Valhalla (1895) by Lorenz Frølich
Valhalla is referenced at length in the Poetic Edda poem Grímnismál, and Helgakviða Hundingsbana II, while Valhalla receives lesser direct references in stanza 32 of the Völuspá, where the god Baldr's death is referred to as the "woe of Valhalla", and in stanzas 1 to 3 of Hyndluljóð, where the goddess Freyja states her intention of riding to Valhalla with Hyndla, in an effort to help Óttar, as well as in stanzas 6 through 7, where Valhalla is mentioned again during a dispute between the two.
Grímnismál
In stanzas 8 to 10 of Grímnismál, the god Odin (in the guise of Grímnir) proclaims Valhalla is in the realm of Glaðsheimr.
Odin describes Valhalla as shining and golden, and it "rises
peacefully" as seen from afar. From Valhalla, every day Odin chooses
from those killed in combat. Valhalla has spear-shafts for rafters, a roof thatched with shields, coats of mail are strewn over its benches, a wolf hangs in front of its west doors, and an eagle hovers above it.
From stanzas 22 to 24, more details are given by Odin about
Valhalla: the holy doors of the ancient gate Valgrind stand before
Valhalla, Valhalla has five hundred and forty doors so eight hundred men can pass through simultaneously (from which the einherjar will flow forth to engage the wolf Fenrir at Ragnarök). Within Valhalla exists Thor's hall Bilskirnir,
and within it exist five hundred and forty rooms, and of all the halls
within Valhalla, Odin states he thinks his son's may be greatest. In stanzas 25 through 26, Odin states the goat Heiðrún and the hartEikþyrnir stand on top of Valhalla and graze on the branches of the tree Læraðr. Heiðrún’s udder produces vats of mead, a liquor beyond compare, and Eikþyrnir's antlers drip liquid into the spring Hvergelmir from which flows forth all waters.
So was Helgi beside the chieftains
like the bright-growing ash beside the thorn-bush
and the young stag, drenched in dew,
who surpasses all other animals
and whose horns glow against the sky itself.
Prose follows after this stanza, stating a burial-mound
was made for Helgi. After Helgi arrived in Valhalla, he was asked by
Odin to manage things with him. In stanza 39, Helgi, now in Valhalla,
has his former enemy Hunding—also in Valhalla—do menial tasks; fetching
foot-baths for all of the men there, kindling fire, tying dogs, keeping
watch of horses, and feeding the pigs before he can get any sleep. In
stanzas 40 to 42, Helgi returns to Midgard from Valhalla with a host of
men. An unnamed maid of Sigrún, Helgi's valkyrie wife, sees Helgi and
his large host of men riding into the mound. The maid asks if she is
experiencing a delusion, if Ragnarök is started, or if Helgi and his men
were allowed to return.
In the following stanzas, Helgi responds none of these things
occurred, and so Sigrún's maid goes home to Sigrún. The maid tells
Sigrún the burial mound is opened, and Sigrún should go to Helgi there.
Helgi asked her to come and tend his wounds after they opened and are
bleeding. Sigrún goes into the mound, and finds Helgi is drenched in
gore, his hair is thick with frost. Filled with joy at the re-union,
Sigrún kisses him before he can remove his coat of mail, and asks how
she can heal him. Sigrún makes a bed there, and the two sleep together
in the enclosed burial mound. Helgi awakens, stating he must "ride along
the blood-red roads, to set the pale horse to tread the path of the
sky," and return before the rooster Salgófnir crows. Helgi and the host
of men ride away, and Sigrún and her servant go back to their house.
Sigrún orders her maid to wait for him by the mound the next night, but
after she arrives at dawn, she finds he is still journeying. The prose
narrative at the end of the poem relates Sigrún dies of sadness, but the
two are thought to be re-born as Helgi Haddingjaskati and the valkyrie Kára.
Valhalla is first mentioned in chapter 2 of the Prose Edda book Gylfaginning, where it is described partially in euhemerized form. In the chapter, King Gylfi sets out to Asgard in the guise of an old man going by the name of Gangleri to find the source of the power of the gods.
The narrative states the Æsir
prophesied his arrival and prepared grand illusions for him, so as
Gangerli enters the fortress, he sees a hall of such a height, he has
trouble seeing over it, and notices the roof of the hall is covered in
golden shields, as if they were shingles. Snorri quotes a stanza by the
skald Þjóðólfr of Hvinir
(c. 900). As he continues, Gangleri sees a man in the doorway of the
hall juggling short swords, and keeping seven in the air simultaneously.
Among other things, the man says the hall belongs to his king, and adds
he can take Gangleri to the king. Gangleri follows him, and the door
closes behind him. All around him, he sees many living areas, and
throngs of people, some of which are playing games, some are drinking,
and others are fighting with weapons. Gangleri sees three thrones, and
three figures sitting upon them: High sitting on the lowest throne, Just-As-High sitting on the next highest throne, and Third sitting on the highest. The man guiding Gangleri tells him High is the king of the hall.
In chapter 20, Third states Odin mans Valhalla with the Einherjar: those killed in battle and become Odin's adopted sons. In chapter 36, High states valkyries serve drinks and see to the tables in Valhalla, and Grímnismál
stanzas 40 to 41 are quoted in reference to this. High continues the
valkyries are sent by Odin to every battle; they choose who is to die,
and determine victory.
In chapter 38, Gangleri says: "You say all men who have fallen in
battle from the beginning of the world are now with Odin in Valhalla.
With what does he feed them? I should think the crowd there is large."
High responds this is indeed true, a huge amount are already in
Valhalla, but yet this amount will seem to be too few before "the wolf
comes." High describes there are never too many to feed in Valhalla, for
they feast from Sæhrímnir (here described as a boar), and this beast is cooked every day and is again whole every night. Grímnismál
stanza 18 is recounted. Gangleri asks if Odin eats the same food as the
Einherjar, and High responds Odin needs nothing to eat—Odin only
consumes wine—and he gives his food to his wolves Geri and Freki. Grímnismál stanza 19 is recounted. High additionally states, at sunrise, Odin sends his ravens Huginn and Muninn from Valhalla to fly throughout the entire world, and they return in time for the first meal there.
In chapter 39, Gangleri asks about the food and drinks the
Einherjar consume, and asks if only water is available there. High
replies of course, Valhalla has food and drinks fit for kings and jarls,
for the mead consumed in Valhalla is produced from the udders of the
goat Heiðrún, who in turn feeds on the leaves of the "famous tree"
Læraðr. The goat produces so much mead in a day, it fills a massive vat
large enough for all of the Einherjar in Valhalla to satisfy their
thirst from it. High further states the stag Eikþyrnir stands atop
Valhalla and chews on the branches of Læraðr. So much moisture drips
from his horns, it falls down to the well Hvelgelmir, resulting in
numerous rivers.
In chapter 40, Gangleri muses Valhalla must be quite crowded, to
which High responds Valhalla is massive and remains roomy despite the
large amount of inhabitants, and then quotes Grímnismál stanza
23. In chapter 41, Gangleri says Odin seems to be quite a powerful lord,
controlling quite a big army, but he wonders how the Einherjar keep
busy while they are not drinking. High replies daily, after they dressed
and put on their war gear, they go out to the courtyard and battle
one-on-one combat for sport. Then, before mealtime, they ride home to
Valhalla and drink. High quotes Vafþrúðnismál
stanza 41. In chapter 42, High describes "right at the beginning, while
the gods were settling", they established Asgard, then built Valhalla. The death of the god Baldr is recounted in chapter 49, with the mistletoe used to kill Baldr is described as growing west of Valhalla.
Skáldskaparmál
At the beginning of Skáldskaparmál, a partially euhemerized account is given of Ægir
visiting the gods in Asgard and shimmering swords are brought out and
used as their sole source of light as they drink. There, numerous gods
feast, they have plenty of strong mead, and the hall has wall-panels covered with attractive shields. This location is confirmed as Valhalla in chapter 33.
In chapter 2, a quote from the anonymous 10th century poem Eiríksmál is provided (see the Fagrskinna section below for more detail and another translation from another source):
What sort of dream is that, Odin? I dreamed I rose up
before dawn to clear up Val-hall for slain people. I aroused the
Einheriar, bade them get up to strew the benches, clean the beer-cups,
the valkyries to serve wine for the arrival of a prince.
In chapter 17 of Skáldskaparmál, the jötunnHrungnir is in a rage and, while attempting to catch up and attack Odin on his steed Sleipnir,
ends up at the doors to Valhalla. There, the Æsir invite him in for a
drink. Hrungnir goes in, demands a drink, and becomes drunk and
belligerent, stating that he will remove Valhalla and take it to the
land of the jötunn, Jötunheimr,
among various other things. Eventually, the gods tire of his boasting
and invoke Thor, who arrives. Hrungnir states that he is under the
Aesir's protection as a guest and therefore he can't be harmed while in
Valhalla. After an exchange of words, Hrungnir challenges Thor to a duel
at the location of Griotunagardar, resulting in Hrungnir's death.
In chapter 34, the tree Glasir
is stated as located in front of the doors of Valhalla. The tree is
described as having foliage of red gold and being the most beautiful
tree among both gods and men. A quote from a work by the 9th century
skald Bragi Boddason is presented that confirms the description.
Heimskringla
Valhalla is mentioned in euhemerized form and as an element of remaining Norse pagan belief in Heimskringla. In chapter 8 of Ynglinga saga,
the "historical" Odin is described as ordaining burial laws over his
country. These laws include that all the dead are to be burned on a pyre
on a burial mound with their possessions, and their ashes are to be
brought out to sea or buried in the earth. The dead would then arrive in
Valhalla with everything that one had on their pyre, and whatever one
had hidden in the ground. Valhalla is additionally referenced in the phrase "visiting Odin" in a work by the 10th century skald Þjóðólfr of Hvinir describing that, upon his death, King Vanlandi went to Valhalla.
In chapter 32 of Hákonar saga Góða, Haakon I of Norway is given a pagan burial, which is described as sending him on his way to Valhalla. Verses from Hákonarmál are then quoted in support, themselves containing references to Valhalla.
Fagrskinna
In chapter 8 of Fagrskinna a prose narrative states that after the death of her husband Eric Bloodaxe, Gunnhild Mother of Kings had a poem composed about him. The composition is by an anonymous author from the 10th century and is referred to as Eiríksmál, and describes Eric Bloodaxe and five other kings arriving in Valhalla after their death. The poem begins with comments by Odin (as Old Norse Óðinn):
"What kind of a dream is it," said Óðinn,
in which just before daybreak,
I thought I cleared Valhǫll,
for coming of slain men?
I waked the Einherjar,
bade valkyries rise up,
to strew the bench,
and scour the beakers,
wine to carry,
as for a king's coming,
here to me I expect
heroes' coming from the world,
certain great ones,
so glad is my heart.
The god Bragi
asks where a thundering sound is coming from, and says that the benches
of Valhalla are creaking—as if the god Baldr had returned to
Valhalla—and that it sounds like the movement of a thousand. Odin
responds that Bragi knows well that the sounds are for Eric Bloodaxe,
who will soon arrive in Valhalla. Odin tells the heroes Sigmund and Sinfjötli to rise to greet Eric and invite him into the hall, if it is indeed he.
Sigmund asks Odin why he would expect Eric more than any other
king, to which Odin responds that Eric has reddened his gore-drenched
sword with many other lands. Eric arrives, and Sigmund greets him, tells
him that he is welcome to come into the hall, and asks him what other
lords he has brought with him to Valhalla. Eric says that with him are
five kings, that he will tell them the name of them all, and that he,
himself, is the sixth.
References to Valhalla appear in literature, art, and other forms
of media. Examples include K. Ehrenberg's charcoal illustration Gastmahl in Walhalla (mit einziehenden Einheriern) (1880), Richard Wagner's depiction of Valhalla in his opera cycle Der Ring des Nibelungen (1848–1874), the Munich, Germany-based Germanic Neopagan magazine Walhalla (1905–1913), and the comic series Valhalla (1978–2009) by Peter Madsen, and its subsequent animated film of the same name (1986). Valhalla also gives its name to a thrill ride at Blackpool Pleasure Beach, UK.
Before Hunter S. Thompson became the counter-culture's Gonzo journalist, he lived in Big Sur, California, while writing his novel The Rum Diary.
He wrote "Big Sur is very like Valhalla—a place that a lot of people
have heard of, and that very few can tell you anything about" (Proud Highway: Saga of a Desperate Southern Gentleman, chapter 20).
In the 2015 film Mad Max: Fury Road, the cult of the War Boys believe a heroic death in the service of dictator Immortan Joe will take them to Valhalla.
A video game with the title Assassin's Creed Valhalla was released in November 2020. The video game Apex Legends
features a character named Bloodhound, who often references Valhalla
and the Allfather, a commonly used kenning for the Norse god Odin.
Elton John's first album, Empty Sky (1969), contains a song called "Valhalla". Led Zeppelin's "Immigrant Song" from their third album, Led Zeppelin III (1970), contains the following Valhalla reference: "To fight the horde, sing and cry: Valhalla, I am coming."[ Australian band Skeggs's third album, Rehearsal (2021), contains a song called "Valhalla".
In 2020's Red, White, and the Blues: A Long and Hard Ride over Treacherous Terrain, John R. Hall (author, blogger, magician) uses Valhalla
as a literary vehicle numerous times within the book's text. His most
notable usages occurs in the book's front matter and back matter. In
"Sursum Corda" (the book's preface), Hall writes: "I tried my best to
capture the nearly unexplainable, Zen-like state of being I experienced
while on my motorcycle. All I can offer the reader is I glimpsed
Valhalla, and Odin assured me I shall dine with him or Freyja after my
internal combat ends and this soldier's body is put down." On the book's final text page (before the notes section), Hall states: "Valhalla! 'A man can die but once' (Henry IV, Part 2). I am ready ... now all family business is settled."
Thermodynamic equilibrium is an axiomatic concept of thermodynamics. It is an internal state of a single thermodynamic system, or a relation between several thermodynamic systems connected by more or less permeable or impermeable walls. In thermodynamic equilibrium there are no net macroscopicflows of matter or of energy, within a system or between systems. In a system that is in its own state of internal thermodynamic equilibrium, no macroscopic change occurs.
Systems in mutual thermodynamic equilibrium are simultaneously in mutual thermal, mechanical, chemical, and radiative
equilibria. Systems can be in one kind of mutual equilibrium, while not
in others. In thermodynamic equilibrium, all kinds of equilibrium hold
at once and indefinitely, until disturbed by a thermodynamic operation.
In a macroscopic equilibrium, perfectly or almost perfectly balanced
microscopic exchanges occur; this is the physical explanation of the
notion of macroscopic equilibrium.
A thermodynamic system in a state of internal thermodynamic equilibrium has a spatially uniform temperature. Its intensive properties,
other than temperature, may be driven to spatial inhomogeneity by an
unchanging long-range force field imposed on it by its surroundings.
In systems that are at a state of non-equilibrium
there are, by contrast, net flows of matter or energy. If such changes
can be triggered to occur in a system in which they are not already
occurring, the system is said to be in a meta-stable equilibrium.
Though not a widely named "law," it is an axiom of thermodynamics that there exist states of thermodynamic equilibrium. The second law of thermodynamics states that when an isolated
body of material starts from an equilibrium state, in which, portions
of it are held at different states by more or less permeable or
impermeable partitions, and a thermodynamic operation removes or makes
the partitions more permeable, then it spontaneously reaches its own,
new state of internal thermodynamic equilibrium, and this is accompanied
by an increase in the sum of the entropies of the portions.
Classical thermodynamics deals with states of dynamic equilibrium. The state of a system at thermodynamic equilibrium is the one for which some thermodynamic potential is minimized (in the absence of an applied voltage), or for which the entropy (S) is maximized, for specified conditions. One such potential is the Helmholtz free energy (A), for a closed system at constant volume and temperature (controlled by a heat bath):
Another potential, the Gibbs free energy (G),
is minimized at thermodynamic equilibrium in a closed system at
constant temperature and pressure, both controlled by the surroundings:
where T denotes the absolute thermodynamic temperature, P the pressure, S the entropy, V the volume, and U the internal energy of the system. In other words, is a necessary condition for chemical equilibrium under these conditions (in the absence of an applied voltage).
Thermodynamic equilibrium is the unique stable stationary state
that is approached or eventually reached as the system interacts with
its surroundings over a long time. The above-mentioned potentials are
mathematically constructed to be the thermodynamic quantities that are
minimized under the particular conditions in the specified surroundings.
Conditions
For a completely isolated system, S is maximum at thermodynamic equilibrium.
For a closed system at controlled constant temperature and volume, A is minimum at thermodynamic equilibrium.
For a closed system at controlled constant temperature and pressure without an applied voltage, G is minimum at thermodynamic equilibrium.
The various types of equilibriums are achieved as follows:
Two systems are in thermal equilibrium when their temperatures are the same.
Two systems are in mechanical equilibrium when their pressures are the same.
Two systems are in diffusive equilibrium when their chemical potentials are the same.
All forces are balanced and there is no significant external driving force.
Relation of exchange equilibrium between systems
Often
the surroundings of a thermodynamic system may also be regarded as
another thermodynamic system. In this view, one may consider the system
and its surroundings as two systems in mutual contact, with long-range
forces also linking them. The enclosure of the system is the surface of
contiguity or boundary between the two systems. In the thermodynamic
formalism, that surface is regarded as having specific properties of
permeability. For example, the surface of contiguity may be supposed to
be permeable only to heat, allowing energy to transfer only as heat.
Then the two systems are said to be in thermal equilibrium when the
long-range forces are unchanging in time and the transfer of energy as
heat between them has slowed and eventually stopped permanently; this is
an example of a contact equilibrium. Other kinds of contact equilibrium
are defined by other kinds of specific permeability.[2]
When two systems are in contact equilibrium with respect to a
particular kind of permeability, they have common values of the
intensive variable that belongs to that particular kind of permeability.
Examples of such intensive variables are temperature, pressure,
chemical potential.
A contact equilibrium may be regarded also as an exchange
equilibrium. There is a zero balance of rate of transfer of some
quantity between the two systems in contact equilibrium. For example,
for a wall permeable only to heat, the rates of diffusion of internal
energy as heat between the two systems are equal and opposite. An
adiabatic wall between the two systems is 'permeable' only to energy
transferred as work; at mechanical equilibrium the rates of transfer of
energy as work between them are equal and opposite. If the wall is a
simple wall, then the rates of transfer of volume across it are also
equal and opposite; and the pressures on either side of it are equal. If
the adiabatic wall is more complicated, with a sort of leverage, having
an area-ratio, then the pressures of the two systems in exchange
equilibrium are in the inverse ratio of the volume exchange ratio; this
keeps the zero balance of rates of transfer as work.
A radiative exchange can occur between two otherwise separate
systems. Radiative exchange equilibrium prevails when the two systems
have the same temperature.
Thermodynamic state of internal equilibrium of a system
A collection of matter may be entirely isolated
from its surroundings. If it has been left undisturbed for an
indefinitely long time, classical thermodynamics postulates that it is
in a state in which no changes occur within it, and there are no flows
within it. This is a thermodynamic state of internal equilibrium. (This postulate is sometimes, but not often, called the "minus first" law of thermodynamics. One textbook calls it the "zeroth law", remarking that the authors think this more befitting that title than its more customary definition, which apparently was suggested by Fowler.)
Such states are a principal concern in what is known as classical
or equilibrium thermodynamics, for they are the only states of the
system that are regarded as well defined in that subject. A system in
contact equilibrium with another system can by a thermodynamic operation
be isolated, and upon the event of isolation, no change occurs in it. A
system in a relation of contact equilibrium with another system may
thus also be regarded as being in its own state of internal
thermodynamic equilibrium.
Multiple contact equilibrium
The
thermodynamic formalism allows that a system may have contact with
several other systems at once, which may or may not also have mutual
contact, the contacts having respectively different permeabilities. If
these systems are all jointly isolated from the rest of the world those
of them that are in contact then reach respective contact equilibria
with one another.
If several systems are free of adiabatic walls between each
other, but are jointly isolated from the rest of the world, then they
reach a state of multiple contact equilibrium, and they have a common
temperature, a total internal energy, and a total entropy.
Amongst intensive variables, this is a unique property of temperature.
It holds even in the presence of long-range forces. (That is, there is
no "force" that can maintain temperature discrepancies.) For example, in
a system in thermodynamic equilibrium in a vertical gravitational
field, the pressure on the top wall is less than that on the bottom
wall, but the temperature is the same everywhere.
A thermodynamic operation may occur as an event restricted to the
walls that are within the surroundings, directly affecting neither the
walls of contact of the system of interest with its surroundings, nor
its interior, and occurring within a definitely limited time. For
example, an immovable adiabatic wall may be placed or removed within the
surroundings. Consequent upon such an operation restricted to the
surroundings, the system may be for a time driven away from its own
initial internal state of thermodynamic equilibrium. Then, according to
the second law of thermodynamics, the whole undergoes changes and
eventually reaches a new and final equilibrium with the surroundings.
Following Planck, this consequent train of events is called a natural thermodynamic process.
It is allowed in equilibrium thermodynamics just because the initial
and final states are of thermodynamic equilibrium, even though during
the process there is transient departure from thermodynamic equilibrium,
when neither the system nor its surroundings are in well defined states
of internal equilibrium. A natural process proceeds at a finite rate
for the main part of its course. It is thereby radically different from a
fictive quasi-static 'process' that proceeds infinitely slowly
throughout its course, and is fictively 'reversible'. Classical
thermodynamics allows that even though a process may take a very long
time to settle to thermodynamic equilibrium, if the main part of its
course is at a finite rate, then it is considered to be natural, and to
be subject to the second law of thermodynamics, and thereby
irreversible. Engineered machines and artificial devices and
manipulations are permitted within the surroundings.
The allowance of such operations and devices in the surroundings but
not in the system is the reason why Kelvin in one of his statements of
the second law of thermodynamics spoke of "inanimate" agency; a system in thermodynamic equilibrium is inanimate.
Otherwise, a thermodynamic operation may directly affect a wall of the system.
It is often convenient to suppose that some of the surrounding
subsystems are so much larger than the system that the process can
affect the intensive variables only of the surrounding subsystems, and
they are then called reservoirs for relevant intensive variables.
Local and global equilibrium
It
is useful to distinguish between global and local thermodynamic
equilibrium. In thermodynamics, exchanges within a system and between
the system and the outside are controlled by intensive parameters. As an example, temperature controls heat exchanges. Global thermodynamic equilibrium (GTE) means that those intensive parameters are homogeneous throughout the whole system, while local thermodynamic equilibrium
(LTE) means that those intensive parameters are varying in space and
time, but are varying so slowly that, for any point, one can assume
thermodynamic equilibrium in some neighborhood about that point.
If the description of the system requires variations in the
intensive parameters that are too large, the very assumptions upon which
the definitions of these intensive parameters are based will break
down, and the system will be in neither global nor local equilibrium.
For example, it takes a certain number of collisions for a particle to
equilibrate to its surroundings. If the average distance it has moved
during these collisions removes it from the neighborhood it is
equilibrating to, it will never equilibrate, and there will be no LTE.
Temperature is, by definition, proportional to the average internal
energy of an equilibrated neighborhood. Since there is no equilibrated
neighborhood, the concept of temperature doesn't hold, and the
temperature becomes undefined.
It is important to note that this local equilibrium may apply
only to a certain subset of particles in the system. For example, LTE is
usually applied only to massive particles. In a radiating gas, the photons
being emitted and absorbed by the gas doesn't need to be in a
thermodynamic equilibrium with each other or with the massive particles
of the gas in order for LTE to exist. In some cases, it is not
considered necessary for free electrons to be in equilibrium with the
much more massive atoms or molecules for LTE to exist.
As an example, LTE will exist in a glass of water that contains a melting ice cube.
The temperature inside the glass can be defined at any point, but it is
colder near the ice cube than far away from it. If energies of the
molecules located near a given point are observed, they will be
distributed according to the Maxwell–Boltzmann distribution
for a certain temperature. If the energies of the molecules located
near another point are observed, they will be distributed according to
the Maxwell–Boltzmann distribution for another temperature.
Local thermodynamic equilibrium does not require either local or
global stationarity. In other words, each small locality need not have a
constant temperature. However, it does require that each small locality
change slowly enough to practically sustain its local Maxwell–Boltzmann
distribution of molecular velocities. A global non-equilibrium state
can be stably stationary only if it is maintained by exchanges between
the system and the outside. For example, a globally-stable stationary
state could be maintained inside the glass of water by continuously
adding finely powdered ice into it in order to compensate for the
melting, and continuously draining off the meltwater. Natural transport phenomena may lead a system from local to global thermodynamic equilibrium. Going back to our example, the diffusion
of heat will lead our glass of water toward global thermodynamic
equilibrium, a state in which the temperature of the glass is completely
homogeneous.
Reservations
Careful
and well informed writers about thermodynamics, in their accounts of
thermodynamic equilibrium, often enough make provisos or reservations to
their statements. Some writers leave such reservations merely implied
or more or less unstated.
For example, one widely cited writer, H. B. Callen
writes in this context: "In actuality, few systems are in absolute and
true equilibrium." He refers to radioactive processes and remarks that
they may take "cosmic times to complete, [and] generally can be
ignored". He adds "In practice, the criterion for equilibrium is
circular. Operationally, a system is in an equilibrium state if its properties are consistently described by thermodynamic theory!"
J.A. Beattie and I. Oppenheim write: "Insistence on a strict
interpretation of the definition of equilibrium would rule out the
application of thermodynamics to practically all states of real
systems."
Another author, cited by Callen as giving a "scholarly and rigorous treatment", and cited by Adkins as having written a "classic text", A.B. Pippard
writes in that text: "Given long enough a supercooled vapour will
eventually condense, ... . The time involved may be so enormous,
however, perhaps 10100 years or more, ... . For most
purposes, provided the rapid change is not artificially stimulated, the
systems may be regarded as being in equilibrium."
Another author, A. Münster, writes in this context. He observes
that thermonuclear processes often occur so slowly that they can be
ignored in thermodynamics. He comments: "The concept 'absolute
equilibrium' or 'equilibrium with respect to all imaginable processes',
has therefore, no physical significance." He therefore states that: "...
we can consider an equilibrium only with respect to specified processes
and defined experimental conditions."
According to L. Tisza:
"... in the discussion of phenomena near absolute zero. The absolute
predictions of the classical theory become particularly vague because
the occurrence of frozen-in nonequilibrium states is very common."
Definitions
The
most general kind of thermodynamic equilibrium of a system is through
contact with the surroundings that allows simultaneous passages of all
chemical substances and all kinds of energy. A system in thermodynamic
equilibrium may move with uniform acceleration through space but must
not change its shape or size while doing so; thus it is defined by a
rigid volume in space. It may lie within external fields of force,
determined by external factors of far greater extent than the system
itself, so that events within the system cannot in an appreciable amount
affect the external fields of force. The system can be in thermodynamic
equilibrium only if the external force fields are uniform, and are
determining its uniform acceleration, or if it lies in a non-uniform
force field but is held stationary there by local forces, such as
mechanical pressures, on its surface.
Thermodynamic equilibrium is a primitive notion of the theory of thermodynamics. According to P.M. Morse:
"It should be emphasized that the fact that there are thermodynamic
states, ..., and the fact that there are thermodynamic variables which
are uniquely specified by the equilibrium state ... are not
conclusions deduced logically from some philosophical first principles.
They are conclusions ineluctably drawn from more than two centuries of
experiments."
This means that thermodynamic equilibrium is not to be defined solely
in terms of other theoretical concepts of thermodynamics. M. Bailyn
proposes a fundamental law of thermodynamics that defines and postulates
the existence of states of thermodynamic equilibrium.
Textbook definitions of thermodynamic equilibrium are often stated carefully, with some reservation or other.
For example, A. Münster writes: "An isolated system is in
thermodynamic equilibrium when, in the system, no changes of state are
occurring at a measurable rate." There are two reservations stated here;
the system is isolated; any changes of state are immeasurably slow. He
discusses the second proviso by giving an account of a mixture oxygen
and hydrogen at room temperature in the absence of a catalyst. Münster
points out that a thermodynamic equilibrium state is described by fewer
macroscopic variables than is any other state of a given system. This is
partly, but not entirely, because all flows within and through the
system are zero.
R. Haase's presentation of thermodynamics does not start with a
restriction to thermodynamic equilibrium because he intends to allow for
non-equilibrium thermodynamics. He considers an arbitrary system with
time invariant properties. He tests it for thermodynamic equilibrium by
cutting it off from all external influences, except external force
fields. If after insulation, nothing changes, he says that the system
was in equilibrium.
In a section headed "Thermodynamic equilibrium", H.B. Callen
defines equilibrium states in a paragraph. He points out that they "are
determined by intrinsic factors" within the system. They are "terminal
states", towards which the systems evolve, over time, which may occur
with "glacial slowness".
This statement does not explicitly say that for thermodynamic
equilibrium, the system must be isolated; Callen does not spell out what
he means by the words "intrinsic factors".
Another textbook writer, C.J. Adkins, explicitly allows
thermodynamic equilibrium to occur in a system which is not isolated.
His system is, however, closed with respect to transfer of matter. He
writes: "In general, the approach to thermodynamic equilibrium will
involve both thermal and work-like interactions with the surroundings."
He distinguishes such thermodynamic equilibrium from thermal
equilibrium, in which only thermal contact is mediating transfer of
energy.
Another textbook author, J.R. Partington, writes: "(i) An equilibrium state is one which is independent of time."
But, referring to systems "which are only apparently in equilibrium",
he adds : "Such systems are in states of ″false equilibrium.″"
Partington's statement does not explicitly state that the equilibrium
refers to an isolated system. Like Münster, Partington also refers to
the mixture of oxygen and hydrogen. He adds a proviso that "In a true
equilibrium state, the smallest change of any external condition which
influences the state will produce a small change of state ..."
This proviso means that thermodynamic equilibrium must be stable
against small perturbations; this requirement is essential for the
strict meaning of thermodynamic equilibrium.
A student textbook by F.H. Crawford has a section headed
"Thermodynamic Equilibrium". It distinguishes several drivers of flows,
and then says: "These are examples of the apparently universal tendency
of isolated systems toward a state of complete mechanical, thermal,
chemical, and electrical—or, in a single word, thermodynamic—equilibrium."
A monograph on classical thermodynamics by H.A. Buchdahl
considers the "equilibrium of a thermodynamic system", without actually
writing the phrase "thermodynamic equilibrium". Referring to systems
closed to exchange of matter, Buchdahl writes: "If a system is in a
terminal condition which is properly static, it will be said to be in equilibrium."
Buchdahl's monograph also discusses amorphous glass, for the purposes
of thermodynamic description. It states: "More precisely, the glass may
be regarded as being in equilibrium so long as experimental tests show that 'slow' transitions are in effect reversible."
It is not customary to make this proviso part of the definition of
thermodynamic equilibrium, but the converse is usually assumed: that if a
body in thermodynamic equilibrium is subject to a sufficiently slow
process, that process may be considered to be sufficiently nearly
reversible, and the body remains sufficiently nearly in thermodynamic
equilibrium during the process.
A. Münster carefully extends his definition of thermodynamic equilibrium for isolated systems by introducing a concept of contact equilibrium.
This specifies particular processes that are allowed when considering
thermodynamic equilibrium for non-isolated systems, with special concern
for open systems, which may gain or lose matter from or to their
surroundings. A contact equilibrium is between the system of interest
and a system in the surroundings, brought into contact with the system
of interest, the contact being through a special kind of wall; for the
rest, the whole joint system is isolated. Walls of this special kind
were also considered by C. Carathéodory,
and are mentioned by other writers also. They are selectively
permeable. They may be permeable only to mechanical work, or only to
heat, or only to some particular chemical substance. Each contact
equilibrium defines an intensive parameter; for example, a wall
permeable only to heat defines an empirical temperature. A contact
equilibrium can exist for each chemical constituent of the system of
interest. In a contact equilibrium, despite the possible exchange
through the selectively permeable wall, the system of interest is
changeless, as if it were in isolated thermodynamic equilibrium. This
scheme follows the general rule that "... we can consider an equilibrium
only with respect to specified processes and defined experimental
conditions."
Thermodynamic equilibrium for an open system means that, with respect
to every relevant kind of selectively permeable wall, contact
equilibrium exists when the respective intensive parameters of the
system and surroundings are equal.
This definition does not consider the most general kind of
thermodynamic equilibrium, which is through unselective contacts. This
definition does not simply state that no current of matter or energy
exists in the interior or at the boundaries; but it is compatible with
the following definition, which does so state.
M. Zemansky
also distinguishes mechanical, chemical, and thermal equilibrium. He
then writes: "When the conditions for all three types of equilibrium are
satisfied, the system is said to be in a state of thermodynamic
equilibrium".
P.M. Morse writes that thermodynamics is concerned with "states of thermodynamic equilibrium".
He also uses the phrase "thermal equilibrium" while discussing transfer
of energy as heat between a body and a heat reservoir in its
surroundings, though not explicitly defining a special term 'thermal
equilibrium'.
J.R. Waldram writes of "a definite thermodynamic state". He
defines the term "thermal equilibrium" for a system "when its
observables have ceased to change over time". But shortly below that
definition he writes of a piece of glass that has not yet reached its "full thermodynamic equilibrium state".
Considering equilibrium states, M. Bailyn writes: "Each intensive
variable has its own type of equilibrium." He then defines thermal
equilibrium, mechanical equilibrium, and material equilibrium.
Accordingly, he writes: "If all the intensive variables become uniform, thermodynamic equilibrium is said to exist." He is not here considering the presence of an external force field.
J.G. Kirkwood and I. Oppenheim define thermodynamic equilibrium as follows: "A system is in a state of thermodynamic equilibrium
if, during the time period allotted for experimentation, (a) its
intensive properties are independent of time and (b) no current of
matter or energy exists in its interior or at its boundaries with the
surroundings." It is evident that they are not restricting the
definition to isolated or to closed systems. They do not discuss the
possibility of changes that occur with "glacial slowness", and proceed
beyond the time period allotted for experimentation. They note that for
two systems in contact, there exists a small subclass of intensive
properties such that if all those of that small subclass are
respectively equal, then all respective intensive properties are equal.
States of thermodynamic equilibrium may be defined by this subclass,
provided some other conditions are satisfied.
Characteristics of a state of internal thermodynamic equilibrium
Homogeneity in the absence of external forces
A
thermodynamic system consisting of a single phase in the absence of
external forces, in its own internal thermodynamic equilibrium, is
homogeneous.
This means that the material in any small volume element of the system
can be interchanged with the material of any other geometrically
congruent volume element of the system, and the effect is to leave the
system thermodynamically unchanged. In general, a strong external force
field makes a system of a single phase in its own internal thermodynamic
equilibrium inhomogeneous with respect to some intensive variables. For example, a relatively dense component of a mixture can be concentrated by centrifugation.
Uniform temperature
Such equilibrium inhomogeneity, induced by external forces, does not occur for the intensive variable temperature. According to E.A. Guggenheim, "The most important conception of thermodynamics is temperature."
Planck introduces his treatise with a brief account of heat and
temperature and thermal equilibrium, and then announces: "In the
following we shall deal chiefly with homogeneous, isotropic bodies of
any form, possessing throughout their substance the same temperature and
density, and subject to a uniform pressure acting everywhere
perpendicular to the surface."
As did Carathéodory, Planck was setting aside surface effects and
external fields and anisotropic crystals. Though referring to
temperature, Planck did not there explicitly refer to the concept of
thermodynamic equilibrium. In contrast, Carathéodory's scheme of
presentation of classical thermodynamics for closed systems postulates
the concept of an "equilibrium state" following Gibbs (Gibbs speaks
routinely of a "thermodynamic state"), though not explicitly using the
phrase 'thermodynamic equilibrium', nor explicitly postulating the
existence of a temperature to define it.
The temperature within a system in thermodynamic equilibrium is
uniform in space as well as in time. In a system in its own state of
internal thermodynamic equilibrium, there are no net internal
macroscopic flows. In particular, this means that all local parts of the
system are in mutual radiative exchange equilibrium. This means that
the temperature of the system is spatially uniform.
This is so in all cases, including those of non-uniform external force
fields. For an externally imposed gravitational field, this may be
proved in macroscopic thermodynamic terms, by the calculus of
variations, using the method of Langrangian multipliers. Considerations of kinetic theory or statistical mechanics also support this statement.
In order that a system may be in its own internal state of
thermodynamic equilibrium, it is of course necessary, but not
sufficient, that it be in its own internal state of thermal equilibrium;
it is possible for a system to reach internal mechanical equilibrium
before it reaches internal thermal equilibrium.
Number of real variables needed for specification
In
his exposition of his scheme of closed system equilibrium
thermodynamics, C. Carathéodory initially postulates that experiment
reveals that a definite number of real variables define the states that
are the points of the manifold of equilibria.
In the words of Prigogine and Defay (1945): "It is a matter of
experience that when we have specified a certain number of macroscopic
properties of a system, then all the other properties are fixed." As noted above, according to A. Münster, the number of variables needed
to define a thermodynamic equilibrium is the least for any state of a
given isolated system. As noted above, J.G. Kirkwood and I. Oppenheim
point out that a state of thermodynamic equilibrium may be defined by a
special subclass of intensive variables, with a definite number of
members in that subclass.
If the thermodynamic equilibrium lies in an external force field,
it is only the temperature that can in general be expected to be
spatially uniform. Intensive variables other than temperature will in
general be non-uniform if the external force field is non-zero. In such a
case, in general, additional variables are needed to describe the
spatial non-uniformity.
Stability against small perturbations
As
noted above, J.R. Partington points out that a state of thermodynamic
equilibrium is stable against small transient perturbations. Without
this condition, in general, experiments intended to study systems in
thermodynamic equilibrium are in severe difficulties.
Approach to thermodynamic equilibrium within an isolated system
When
a body of material starts from a non-equilibrium state of inhomogeneity
or chemical non-equilibrium, and is then isolated, it spontaneously
evolves towards its own internal state of thermodynamic equilibrium. It
is not necessary that all aspects of internal thermodynamic equilibrium
be reached simultaneously; some can be established before others. For
example, in many cases of such evolution, internal mechanical
equilibrium is established much more rapidly than the other aspects of
the eventual thermodynamic equilibrium.
Another example is that, in many cases of such evolution, thermal
equilibrium is reached much more rapidly than chemical equilibrium.
Fluctuations within an isolated system in its own internal thermodynamic equilibrium
In
an isolated system, thermodynamic equilibrium by definition persists
over an indefinitely long time. In classical physics it is often
convenient to ignore the effects of measurement and this is assumed in
the present account.
To consider the notion of fluctuations in an isolated
thermodynamic system, a convenient example is a system specified by its
extensive state variables, internal energy, volume, and mass
composition. By definition they are time-invariant. By definition, they
combine with time-invariant nominal values of their conjugate intensive
functions of state, inverse temperature, pressure divided by
temperature, and the chemical potentials divided by temperature, so as
to exactly obey the laws of thermodynamics.
But the laws of thermodynamics, combined with the values of the
specifying extensive variables of state, are not sufficient to provide
knowledge of those nominal values. Further information is needed,
namely, of the constitutive properties of the system.
It may be admitted that on repeated measurement of those
conjugate intensive functions of state, they are found to have slightly
different values from time to time. Such variability is regarded as due
to internal fluctuations. The different measured values average to their
nominal values.
If the system is truly macroscopic as postulated by classical
thermodynamics, then the fluctuations are too small to detect
macroscopically. This is called the thermodynamic limit. In effect, the
molecular nature of matter and the quantal nature of momentum transfer
have vanished from sight, too small to see. According to Buchdahl: "...
there is no place within the strictly phenomenological theory for the
idea of fluctuations about equilibrium (see, however, Section 76)."
If the system is repeatedly subdivided, eventually a system is
produced that is small enough to exhibit obvious fluctuations. This is a
mesoscopic level of investigation. The fluctuations are then directly
dependent on the natures of the various walls of the system. The precise
choice of independent state variables is then important. At this stage,
statistical features of the laws of thermodynamics become apparent.
If the mesoscopic system is further repeatedly divided,
eventually a microscopic system is produced. Then the molecular
character of matter and the quantal nature of momentum transfer become
important in the processes of fluctuation. One has left the realm of
classical or macroscopic thermodynamics, and one needs quantum
statistical mechanics. The fluctuations can become relatively dominant,
and questions of measurement become important.
The statement that 'the system is its own internal thermodynamic
equilibrium' may be taken to mean that 'indefinitely many such
measurements have been taken from time to time, with no trend in time in
the various measured values'. Thus the statement, that 'a system is in
its own internal thermodynamic equilibrium, with stated nominal values
of its functions of state conjugate to its specifying state variables',
is far far more informative than a statement that 'a set of single
simultaneous measurements of those functions of state have those same
values'. This is because the single measurements might have been made
during a slight fluctuation, away from another set of nominal values of
those conjugate intensive functions of state, that is due to unknown and
different constitutive properties. A single measurement cannot tell
whether that might be so, unless there is also knowledge of the nominal
values that belong to the equilibrium state.
An explicit distinction between 'thermal equilibrium' and
'thermodynamic equilibrium' is made by B. C. Eu. He considers two
systems in thermal contact, one a thermometer, the other a system in
which there are several occurring irreversible processes, entailing
non-zero fluxes; the two systems are separated by a wall permeable only
to heat. He considers the case in which, over the time scale of
interest, it happens that both the thermometer reading and the
irreversible processes are steady. Then there is thermal equilibrium
without thermodynamic equilibrium. Eu proposes consequently that the
zeroth law of thermodynamics can be considered to apply even when
thermodynamic equilibrium is not present; also he proposes that if
changes are occurring so fast that a steady temperature cannot be
defined, then "it is no longer possible to describe the process by means
of a thermodynamic formalism. In other words, thermodynamics has no
meaning for such a process." This illustrates the importance for thermodynamics of the concept of temperature.
Thermal equilibrium is achieved when two systems in thermal contact
with each other cease to have a net exchange of energy. It follows that
if two systems are in thermal equilibrium, then their temperatures are
the same.
Thermal equilibrium occurs when a system's macroscopic thermal observables have ceased to change with time. For example, an ideal gas whose distribution function has stabilised to a specific Maxwell–Boltzmann distribution would be in thermal equilibrium. This outcome allows a single temperature and pressure
to be attributed to the whole system. For an isolated body, it is quite
possible for mechanical equilibrium to be reached before thermal
equilibrium is reached, but eventually, all aspects of equilibrium,
including thermal equilibrium, are necessary for thermodynamic
equilibrium.
A system's internal state of thermodynamic equilibrium should be
distinguished from a "stationary state" in which thermodynamic
parameters are unchanging in time but the system is not isolated, so
that there are, into and out of the system, non-zero macroscopic fluxes
which are constant in time.
Non-equilibrium thermodynamics is a branch of thermodynamics that
deals with systems that are not in thermodynamic equilibrium. Most
systems found in nature are not in thermodynamic equilibrium because
they are changing or can be triggered to change over time, and are
continuously and discontinuously subject to flux of matter and energy to
and from other systems. The thermodynamic study of non-equilibrium
systems requires more general concepts than are dealt with by
equilibrium thermodynamics. Many natural systems still today remain beyond the scope of currently known macroscopic thermodynamic methods.
Laws governing systems which are far from equilibrium are also
debatable. One of the guiding principles for these systems is the
maximum entropy production principle. It states that a non-equilibrium system evolves such as to maximize its entropy production.
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