L-DOPA

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https://en.wikipedia.org/wiki/L-DOPA

l-DOPA

Skeletal formula of L-DOPA

Ball-and-stick model of the zwitterionic form of L-DOPA found in the crystal structure

Chemical and physical data
Formula	C9H11NO4
Molar mass	197.190 g·mol−1

l-DOPA, also known as levodopa and l-3,4-dihydroxyphenylalanine, is an amino acid that is made and used as part of the normal biology of some plants and animals, including humans. Humans, as well as a portion of the other animals that utilize l-DOPA, make it via biosynthesis from the amino acid l-tyrosine. l-DOPA is the precursor to the neurotransmitters dopamine, norepinephrine (noradrenaline), and epinephrine (adrenaline), which are collectively known as catecholamines. Furthermore, l-DOPA itself mediates neurotrophic factor release by the brain and CNS. l-DOPA can be manufactured and in its pure form is sold as a psychoactive drug with the INN levodopa; trade names include Sinemet, Pharmacopa, Atamet, and Stalevo. As a drug, it is used in the clinical treatment of Parkinson's disease and dopamine-responsive dystonia.

l-DOPA has a counterpart with opposite chirality, d-DOPA. As is true for many molecules, the human body produces only one of these isomers (the l-DOPA form). The enantiomeric purity of l-DOPA may be analyzed by determination of the optical rotation or by chiral thin-layer chromatography.

Medical use

l-DOPA crosses the protective blood–brain barrier, whereas dopamine itself cannot. Thus, l-DOPA is used to increase dopamine concentrations in the treatment of Parkinson's disease, Parkinsonism, dopamine-responsive dystonia and Parkinson-plus syndrome. The therapeutic efficacy is different for different kinds of symptoms. Bradykinesia and rigidity are the most responsive symptoms while tremors are less responsive to levodopa therapy. Speech, swallowing disorders, postural instability and freezing gait are the least responsive symptoms.

Once l-DOPA has entered the central nervous system, it is converted into dopamine by the enzyme aromatic l-amino acid decarboxylase, also known as DOPA decarboxylase. Pyridoxal phosphate (vitamin B₆) is a required cofactor in this reaction, and may occasionally be administered along with l-DOPA, usually in the form of pyridoxine. Because levodopa bypasses the enzyme tyrosine hydroxylase, the rate-limiting step in dopamine synthesis, it is much more readily converted to dopamine than tyrosine, which is normally the natural precursor for dopamine production.

In humans, conversion of l-DOPA to dopamine does not only occur within the central nervous system. Cells in the peripheral nervous system perform the same task. Thus administering l-DOPA alone will lead to increased dopamine signaling in the periphery as well. Excessive peripheral dopamine signaling is undesirable as it causes many of the adverse side effects seen with sole L-DOPA administration. To bypass these effects, it is standard clinical practice to coadminister (with l-DOPA) a peripheral DOPA decarboxylase inhibitor (DDCI) such as carbidopa (medicines containing carbidopa, either alone or in combination with l-DOPA, are branded as Lodosyn (Aton Pharma) Sinemet (Merck Sharp & Dohme Limited), Pharmacopa (Jazz Pharmaceuticals), Atamet (UCB), Syndopa and Stalevo (Orion Corporation) or with a benserazide (combination medicines are branded Madopar or Prolopa), to prevent the peripheral synthesis of dopamine from l-DOPA). However, when consumed as a botanical extract, for example from M pruriens supplements, a peripheral DOPA decarboxylase inhibitor is not present.

Inbrija (previously known as CVT-301) is an inhaled powder formulation of levodopa indicated for the intermittent treatment of "off episodes" in patients with Parkinson's disease currently taking carbidopa/levodopa. It was approved by the United States Food and Drug Administration on December 21, 2018, and is marketed by Acorda Therapeutics.

Coadministration of pyridoxine without a DDCI accelerates the peripheral decarboxylation of l-DOPA to such an extent that it negates the effects of l-DOPA administration, a phenomenon that historically caused great confusion.

In addition, l-DOPA, co-administered with a peripheral DDCI, is efficacious for the short-term treatment of restless leg syndrome.

The two types of response seen with administration of l-DOPA are:

• The short-duration response is related to the half-life of the drug.
• The longer-duration response depends on the accumulation of effects over at least two weeks, during which ΔFosB accumulates in nigrostriatal neurons. In the treatment of Parkinson's disease, this response is evident only in early therapy, as the inability of the brain to store dopamine is not yet a concern.

Biological role

Biosynthetic pathways for catecholamines and trace amines in the human brain L-Phenylalanine L-Tyrosine L-DOPA Epinephrine Phenethylamine p-Tyramine Dopamine Norepinephrine N-Methylphenethylamine N-Methyltyramine p-Octopamine Synephrine 3-Methoxytyramine AADC AADC AADC primary pathway PNMT PNMT PNMT PNMT AAAH AAAH brain CYP2D6 minor pathway COMT DBH DBH In humans, catecholamines and phenethylaminergic trace amines are derived from the amino acid L-phenylalanine.

l-DOPA is produced from the amino acid l-tyrosine by the enzyme tyrosine hydroxylase. l-DOPA can act as an l-tyrosine mimetic and be incorporated into proteins by mammalian cells in place of L-tyrosine, generating protease-resistant and aggregate-prone proteins in vitro and may contribute to neurotoxicity with chronic l-DOPA administration. It is also the precursor for the monoamine or catecholamine neurotransmitters dopamine, norepinephrine (noradrenaline), and epinephrine (adrenaline). Dopamine is formed by the decarboxylation of l-DOPA by aromatic l-amino acid decarboxylase (AADC).

l-DOPA can be directly metabolized by catechol-O-methyl transferase to 3-O-methyldopa, and then further to vanillactic acid. This metabolic pathway is nonexistent in the healthy body, but becomes important after peripheral l-DOPA administration in patients with Parkinson's disease or in the rare cases of patients with AADC enzyme deficiency.

l-Phenylalanine, l-tyrosine, and l-DOPA are all precursors to the biological pigment melanin. The enzyme tyrosinase catalyzes the oxidation of l-DOPA to the reactive intermediate dopaquinone, which reacts further, eventually leading to melanin oligomers. In addition, tyrosinase can convert tyrosine directly to l-DOPA in the presence of a reducing agent such as ascorbic acid.

Marine adhesion

l-DOPA is a key compound in the formation of marine adhesive proteins, such as those found in mussels. It is believed to be responsible for the water-resistance and rapid curing abilities of these proteins. l-DOPA may also be used to prevent surfaces from fouling by bonding antifouling polymers to a susceptible substrate. The versatile chemistry of L-DOPA can be exploited in nanotechnology. For example, DOPA-containing self-assembling peptides were found to form functional nanostructures, adhesives and gels.

Side effects and adverse reactions

The side effects of l-DOPA may include:

• Hypertension, especially if the dosage is too high
• Arrhythmias, although these are uncommon
• Nausea, which is often reduced by taking the drug with food, although protein reduces drug absorption. l-DOPA is an amino acid, so protein competitively inhibits l-DOPA absorption.
• Gastrointestinal bleeding
• Disturbed respiration, which is not always harmful, and can actually benefit patients with upper airway obstruction
• Hair loss
• Disorientation and confusion
• Extreme emotional states, particularly anxiety, but also excessive libido
• Vivid dreams or insomnia
• Auditory or visual hallucinations
• Effects on learning; some evidence indicates it improves working memory, while impairing other complex functions
• Somnolence and narcolepsy
• A condition similar to stimulant psychosis

Although many adverse effects are associated with l-DOPA, in particular psychiatric ones, it has fewer than other antiparkinsonian agents, such as anticholinergics and dopamine receptor agonists.

More serious are the effects of chronic l-DOPA administration in the treatment of Parkinson's disease, which include:

• End-of-dose deterioration of function
• "On/off" oscillations
• Freezing during movement
• Dose failure (drug resistance)
• Dyskinesia at peak dose (levodopa-induced dyskinesia)
• Possible dopamine dysregulation: The long-term use of l-DOPA in Parkinson's disease has been linked to the so-called dopamine dysregulation syndrome.

Clinicians try to avoid these side effects and adverse reactions by limiting l-DOPA doses as much as possible until absolutely necessary.

The long term use of L-Dopa increases oxidative stress through monoamine oxidase led enzymatic degradation of synthesized dopamine causing neuronal damage and cytotoxicity. The oxidative stress is caused by the formation of reactive oxygen species (H₂O₂) during the monoamine oxidase led metabolism of dopamine. It is further perpetuated by the richness of Fe²⁺ ions in striatum via the Fenton reaction and intracellular autooxidation. The increased oxidation can potentially cause mutations in DNA due to the formation of 8-oxoguanine, which is capable of pairing with adenosine during mitosis.

History

In work that earned him a Nobel Prize in 2000, Swedish scientist Arvid Carlsson first showed in the 1950s that administering l-DOPA to animals with drug-induced (reserpine) Parkinsonian symptoms caused a reduction in the intensity of the animals' symptoms. In 1960/61 Oleh Hornykiewicz, after discovering greatly reduced levels of dopamine in autopsied brains of patients with Parkinson's disease, published together with the neurologist Walther Birkmayer dramatic therapeutic antiparkinson effects of intravenously administered l-DOPA in patients. This treatment was later extended to manganese poisoning and later Parkinsonism by George Cotzias and his coworkers, who used greatly increased oral doses, for which they won the 1969 Lasker Prize. The neurologist Oliver Sacks describes this treatment in human patients with encephalitis lethargica in his 1973 book Awakenings, upon which the 1990 movie of the same name is based. The first study reporting improvements in patients with Parkinson's disease resulting from treatment with L-dopa was published in 1968.

The 2001 Nobel Prize in Chemistry was also related to l-DOPA: the Nobel Committee awarded one-quarter of the prize to William S. Knowles for his work on chirally catalysed hydrogenation reactions, the most noted example of which was used for the synthesis of l-DOPA.

Synthesis of l-DOPA via hydrogenation with C₂-symmetric diphosphine.

Research

Age-related macular degeneration

In 2015, a retrospective analysis comparing the incidence of age-related macular degeneration (AMD) between patients taking versus not taking l-DOPA found that the drug delayed onset of AMD by around 8 years. The authors state that significant effects were obtained for both dry and wet AMD.

Nova

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Nova

Artist's conception of a white dwarf, right, accreting hydrogen from the Roche lobe of its larger companion star

A nova (PL: novae or novas) is a transient astronomical event that causes the sudden appearance of a bright, apparently "new" star (hence the name "nova", which is Latin for "new") that slowly fades over weeks or months. Causes of the dramatic appearance of a nova vary, depending on the circumstances of the two progenitor stars. All observed novae involve white dwarfs in close binary systems. The main sub-classes of novae are classical novae, recurrent novae (RNe), and dwarf novae. They are all considered to be cataclysmic variable stars.

Classical nova eruptions are the most common type. They are likely created in a close binary star system consisting of a white dwarf and either a main sequence, subgiant, or red giant star. When the orbital period falls in the range of several days to one day, the white dwarf is close enough to its companion star to start drawing accreted matter onto the surface of the white dwarf, which creates a dense but shallow atmosphere. This atmosphere, mostly consisting of hydrogen, is thermally heated by the hot white dwarf and eventually reaches a critical temperature causing ignition of rapid runaway fusion.

The sudden increase in energy expels the atmosphere into interstellar space creating the envelope seen as visible light during the nova event. Such were taken in past centuries to be a new star. A few novae produce short-lived nova remnants, lasting for perhaps several centuries. Recurrent nova processes are the same as the classical nova, except that the fusion ignition may be repetitive because the companion star can again feed the dense atmosphere of the white dwarf.

Novae most often occur in the sky along the path of the Milky Way, especially near the observed Galactic Center in Sagittarius; however, they can appear anywhere in the sky. They occur far more frequently than galactic supernovae, averaging about ten per year in the Milky Way. Most are found telescopically, perhaps only one every 12–18 months reaching naked-eye visibility. Novae reaching first or second magnitude occur only several times per century. The last bright nova was V1369 Centauri reaching 3.3 magnitude on 14 December 2013.

Etymology

During the sixteenth century, astronomer Tycho Brahe observed the supernova SN 1572 in the constellation Cassiopeia. He described it in his book De nova stella (Latin for "concerning the new star"), giving rise to the adoption of the name nova. In this work he argued that a nearby object should be seen to move relative to the fixed stars, and that the nova had to be very far away. Although this event was a supernova and not a nova, the terms were considered interchangeable until the 1930s. After this, novae were classified as classical novae to distinguish them from supernovae, as their causes and energies were thought to be different, based solely in the observational evidence.

Although the term "stella nova" means "new star", novae most often take place as a result of white dwarfs, which are remnants of extremely old stars.

Stellar evolution of novae

Nova Eridani 2009 (apparent magnitude ~8.4)

Evolution of potential novae begins with two main sequence stars in a binary system. One of the two evolves into a red giant, leaving its remnant white dwarf core in orbit with the remaining star. The second star—which may be either a main sequence star or an aging giant—begins to shed its envelope onto its white dwarf companion when it overflows its Roche lobe. As a result, the white dwarf steadily captures matter from the companion's outer atmosphere in an accretion disk, and in turn, the accreted matter falls into the atmosphere. As the white dwarf consists of degenerate matter, the accreted hydrogen does not inflate, but its temperature increases. Runaway fusion occurs when the temperature of this atmospheric layer reaches ~20 million K, initiating nuclear burning, via the CNO cycle.

Hydrogen fusion may occur in a stable manner on the surface of the white dwarf for a narrow range of accretion rates, giving rise to a super soft X-ray source, but for most binary system parameters, the hydrogen burning is unstable thermally and rapidly converts a large amount of the hydrogen into other, heavier chemical elements in a runaway reaction, liberating an enormous amount of energy. This blows the remaining gases away from the surface of the white dwarf surface and produces an extremely bright outburst of light.

The rise to peak brightness may be very rapid, or gradual. This is related to the speed class of the nova; yet after the peak, the brightness declines steadily. The time taken for a nova to decay by around 2 or 3 magnitudes from maximum optical brightness is used for classification, via its speed class. Fast novae typically will take fewer than 25 days to decay by 2 magnitudes, while slow novae will take more than 80 days.

Despite their violence, usually the amount of material ejected in novae is only about 1⁄10,000 of a solar mass, quite small relative to the mass of the white dwarf. Furthermore, only five percent of the accreted mass is fused during the power outburst. Nonetheless, this is enough energy to accelerate nova ejecta to velocities as high as several thousand kilometers per second—higher for fast novae than slow ones—with a concurrent rise in luminosity from a few times solar to 50,000–100,000 times solar. In 2010 scientists using NASA's Fermi Gamma-ray Space Telescope discovered that a nova also can emit gamma-rays (>100 MeV).

Potentially, a white dwarf can generate multiple novae over time as additional hydrogen continues to accrete onto its surface from its companion star. An example is RS Ophiuchi, which is known to have flared seven times (in 1898, 1933, 1958, 1967, 1985, 2006, and 2021). Eventually, the white dwarf could explode as a Type Ia supernova if it approaches the Chandrasekhar limit.

Occasionally, novae are bright enough and close enough to Earth to be conspicuous to the unaided eye. The brightest recent example was Nova Cygni 1975. This nova appeared on 29 August 1975, in the constellation Cygnus about five degrees north of Deneb, and reached magnitude 2.0 (nearly as bright as Deneb). The most recent were V1280 Scorpii, which reached magnitude 3.7 on 17 February 2007, and Nova Delphini 2013. Nova Centauri 2013 was discovered 2 December 2013 and so far, is the brightest nova of this millennium, reaching magnitude 3.3.

Helium novae

A helium nova (undergoing a helium flash) is a proposed category of nova events that lacks hydrogen lines in its spectrum. This may be caused by the explosion of a helium shell on a white dwarf. The theory was first proposed in 1989, and the first candidate helium nova to be observed was V445 Puppis in 2000. Since then, four other novae have been proposed as helium novae.

Occurrence rate and astrophysical significance

Astronomers estimate that the Milky Way experiences roughly 30 to 60 novae per year, but a recent examination has found the likely improved rate of about 50±27. The number of novae discovered in the Milky Way each year is much lower, about 10, probably due to distant novae being obscured by gas and dust absorption. Roughly 25 novae brighter than about the twentieth magnitude are discovered in the Andromeda Galaxy each year and smaller numbers are seen in other nearby galaxies. As of 2019, 407 probable novae are recorded in the Milky Way.

Spectroscopic observation of nova ejecta nebulae has shown that they are enriched in elements such as helium, carbon, nitrogen, oxygen, neon, and magnesium. The contribution of novae to the interstellar medium is not great; novae supply only 1⁄50 as much material to the Galaxy as do supernovae, and only 1⁄200 as much as red giant and supergiant stars. Classical novae explosions are galactic producers of the element lithium.

Observed recurrent novae such as RS Ophiuchi (those with periods on the order of decades) are rare. Astronomers theorize, however, that most, if not all, novae are recurrent, albeit on time scales ranging from 1,000 to 100,000 years. The recurrence interval for a nova is less dependent on the accretion rate of the white dwarf than on its mass; with their powerful gravity, massive white dwarfs require less accretion to fuel an eruption than lower-mass ones. Consequently, the interval is shorter for high-mass white dwarfs.

V Sagittae is unusual in that we can predict now that it will go nova in approximately 2083, plus or minus about 11 years.

Subtypes

Novae are classified according to the light curve development speed, so in

• NA: fast novae, with a rapid brightness increase, followed by a brightness decline of 3 magnitudes—to about 1⁄16 brightness—within 100 days.
• NB: slow novae, with magnitudes of 3, decline in 150 days or more.
• NC: very slow novae, also known as symbiotic novae, staying at maximum light for a decade or more and then fading very slowly.
• NR/RN: recurrent novae, novae with two or more eruptions separated by 10–80 years have been observed.

Remnants

GK Persei: Nova of 1901

Main article: Nova remnant

Some novae leave behind visible nebulosity, material expelled in the nova explosion or in multiple explosions.

Novae as distance indicators

Novae have some promise for use as standard candle measurements of distances. For instance, the distribution of their absolute magnitude is bimodal, with a main peak at magnitude −8.8, and a lesser one at −7.5. Novae also have roughly the same absolute magnitude 15 days after their peak (−5.5). Comparisons of nova-based distance estimates to various nearby galaxies and galaxy clusters with those measured with Cepheid variable stars, have shown them to be of comparable accuracy.

Recurrent nova

A recurrent nova (RNe) is an object that has been seen to experience repeated nova eruptions. as well as several extragalactic ones (in the Andromeda Galaxy (M31) and the Large Magellanic Cloud). One of these extragalactic novae, M31N 2008-12a, erupts as frequently as once every 12 months. The recurrent nova typically brightens by about 8.6 magnitudes, whereas a classic nova may brighten by more than 12 magnitudes. Although it is estimated that as many as a quarter of nova systems experience multiple eruptions, only ten recurrent novae have been observed in the Milky Way. The ten known galactic recurrent novae are listed below.

Full name	Discoverer	Magnitude range	Days to drop 3 magnitudes from peak	Known eruption years	Time span (years)	Years since latest eruption
CI Aquilae	K. Reinmuth	8.6–16.3	40	1917, 1941, 2000	24–59	23
V394 Coronae Australis	L. E. Erro	7.2–19.7	6	1949, 1987	38	36
T Coronae Borealis	J. Birmingham	2.5–10.8	6	1866, 1946	80	77
IM Normae	I. E. Woods	8.5–18.5	70	1920, 2002	≤82	21
RS Ophiuchi	W. Fleming	4.8–11	14	1898, 1907, 1933, 1958, 1967, 1985, 2006, 2021	9–26	1
V2487 Ophiuchi	K. Takamizawa (1998)	9.5–17.5	9	1900, 1998	98	25
T Pyxidis	H. Leavitt	6.4–15.5	62	1890, 1902, 1920, 1944, 1967, 2011	12–44	12
V3890 Sagittarii	H. Dinerstein	8.1–18.4	14	1962, 1990, 2019	28–29	3
U Scorpii	N. R. Pogson	7.5–17.6	2.6	1863, 1906, 1917, 1936, 1979, 1987, 1999, 2010, 2022,	8–43	1
V745 Scorpii	L. Plaut	9.4–19.3	7	1937, 1989, 2014	25–52	9

Extragalactic novae

Nova in Andromeda Galaxy

Novae are relatively common in the Andromeda Galaxy (M31). Approximately several dozen novae (brighter than about apparent magnitude 20) are discovered in M31 each year. The Central Bureau for Astronomical Telegrams (CBAT) tracked novae in M31, M33, and M81.

Grand canonical ensemble

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Grand_canonical_ensemble

In statistical mechanics, the grand canonical ensemble (also known as the macrocanonical ensemble) is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium (thermal and chemical) with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

The thermodynamic variables of the grand canonical ensemble are chemical potential (symbol: µ) and absolute temperature (symbol: T). The ensemble is also dependent on mechanical variables such as volume (symbol: V) which influence the nature of the system's internal states. This ensemble is therefore sometimes called the µVT ensemble, as each of these three quantities are constants of the ensemble.

Basics

In simple terms, the grand canonical ensemble assigns a probability P to each distinct microstate given by the following exponential:

${\displaystyle P=e^{\frac{\mathrm{\Omega }+\mu N-E}{kT}},}$

where N is the number of particles in the microstate and E is the total energy of the microstate. k is Boltzmann's constant.

The number Ω is known as the grand potential and is constant for the ensemble. However, the probabilities and Ω will vary if different µ, V, T are selected. The grand potential Ω serves two roles: to provide a normalization factor for the probability distribution (the probabilities, over the complete set of microstates, must add up to one); and, many important ensemble averages can be directly calculated from the function Ω(µ, V, T).

In the case where more than one kind of particle is allowed to vary in number, the probability expression generalizes to

${\displaystyle P=e^{\frac{\mathrm{\Omega }+{\mu }_1{N}_1+{\mu }_2{N}_2+\dots +{\mu }_s{N}_s-E}{kT}},}$

where µ₁ is the chemical potential for the first kind of particles, N₁ is the number of that kind of particle in the microstate, µ₂ is the chemical potential for the second kind of particles and so on (s is the number of distinct kinds of particles). However, these particle numbers should be defined carefully (see the note on particle number conservation below).

The distribution of the grand canonical ensemble is called generalized Boltzmann distribution by some authors.

Grand ensembles are apt for use when describing systems such as the electrons in a conductor, or the photons in a cavity, where the shape is fixed but the energy and number of particles can easily fluctuate due to contact with a reservoir (e.g., an electrical ground or a dark surface, in these cases). The grand canonical ensemble provides a natural setting for an exact derivation of the Fermi–Dirac statistics or Bose–Einstein statistics for a system of non-interacting quantum particles (see examples below).

Note on formulation

An alternative formulation for the same concept writes the probability as ${\displaystyle P=\frac{1}{\mathcal{Z}}{e}^{(\mu N-E)/(kT)}}$, using the grand partition function ${\displaystyle \mathcal{Z}=e^{-\mathrm{\Omega }/(kT)}}$ rather than the grand potential. The equations in this article (in terms of grand potential) may be restated in terms of the grand partition function by simple mathematical manipulations.

Applicability

The grand canonical ensemble is the ensemble that describes the possible states of an isolated system that is in thermal and chemical equilibrium with a reservoir (the derivation proceeds along lines analogous to the heat bath derivation of the normal canonical ensemble, and can be found in Reif). The grand canonical ensemble applies to systems of any size, small or large; it is only necessary to assume that the reservoir with which it is in contact is much larger (i.e., to take the macroscopic limit).

The condition that the system is isolated is necessary in order to ensure it has well-defined thermodynamic quantities and evolution. In practice, however, it is desirable to apply the grand canonical ensemble to describe systems that are in direct contact with the reservoir, since it is that contact that ensures the equilibrium. The use of the grand canonical ensemble in these cases is usually justified either 1) by assuming that the contact is weak, or 2) by incorporating a part of the reservoir connection into the system under analysis, so that the connection's influence on the region of interest is correctly modeled. Alternatively, theoretical approaches can be used to model the influence of the connection, yielding an open statistical ensemble.

Another case in which the grand canonical ensemble appears is when considering a system that is large and thermodynamic (a system that is "in equilibrium with itself"). Even if the exact conditions of the system do not actually allow for variations in energy or particle number, the grand canonical ensemble can be used to simplify calculations of some thermodynamic properties. The reason for this is that various thermodynamic ensembles (microcanonical, canonical) become equivalent in some aspects to the grand canonical ensemble, once the system is very large. Of course, for small systems, the different ensembles are no longer equivalent even in the mean. As a result, the grand canonical ensemble can be highly inaccurate when applied to small systems of fixed particle number, such as atomic nuclei.

Properties

• Uniqueness: The grand canonical ensemble is uniquely determined for a given system at given temperature and given chemical potentials, and does not depend on arbitrary choices such as choice of coordinate system (classical mechanics) or basis (quantum mechanics). The grand canonical ensemble is the only ensemble with constant ${\displaystyle \mu }$, V, and T that reproduces the fundamental thermodynamic relation.
• Statistical equilibrium (steady state): A grand canonical ensemble does not evolve over time, despite the fact that the underlying system is in constant motion. Indeed, the ensemble is only a function of the conserved quantities of the system (energy and particle numbers).
• Thermal and chemical equilibrium with other systems: Two systems, each described by a grand canonical ensemble of equal temperature and chemical potentials, brought into thermal and chemical contact will remain unchanged, and the resulting combined system will be described by a combined grand canonical ensemble of the same temperature and chemical potentials.
• Maximum entropy: For given mechanical parameters (fixed V), the grand canonical ensemble average of the log-probability ${\displaystyle -⟨\log P⟩}$ (also called the "entropy") is the maximum possible for any ensemble (i.e. probability distribution P) with the same ${\displaystyle ⟨E⟩}$, ${\displaystyle ⟨N_1⟩}$, etc.
• Minimum grand potential: For given mechanical parameters (fixed V) and given values of T, µ₁, …, µ_s, the ensemble average ${\displaystyle ⟨E+kT\log P-{\mu }_1{N}_1-\dots {\mu }_s{N}_s⟩}$ is the lowest possible of any ensemble.

Grand potential, ensemble averages, and exact differentials

The partial derivatives of the function Ω(µ₁, …, µ_s, V, T) give important grand canonical ensemble average quantities:

• the averages of numbers of particles

  ${\displaystyle ⟨N_1⟩=-\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }{\mu }_1},\dots ⟨N_s⟩=-\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }{\mu }_s},}$

• the average pressure
  ${\displaystyle ⟨p⟩=-\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }V},}$
• the Gibbs entropy
  ${\displaystyle S=-k⟨\log P⟩=-\frac{\mathrm{\partial }\mathrm{\Omega }}{\mathrm{\partial }T},}$
• and the average energy
  ${\displaystyle ⟨E⟩=\mathrm{\Omega }+⟨N_1⟩{\mu }_1\dots +⟨N_s⟩{\mu }_s+ST.}$

Exact differential: From the above expressions, it can be seen that the function Ω has the exact differential

${\displaystyle d\mathrm{\Omega }=-SdT-⟨N_1⟩d{\mu }_1\dots -⟨N_s⟩d{\mu }_s-⟨p⟩dV.}$

First law of thermodynamics: Substituting the above relationship for ⟨E⟩ into the exact differential of Ω, an equation similar to the first law of thermodynamics is found, except with average signs on some of the quantities:

${\displaystyle d⟨E⟩=TdS+{\mu }_{1}d⟨N_1⟩\dots +{\mu }_{s}d⟨N_s⟩-⟨p⟩dV.}$

Thermodynamic fluctuations: The variances in energy and particle numbers are

${\displaystyle ⟨E^2⟩-⟨E{⟩}^2=k{T}^2\frac{\mathrm{\partial }⟨E⟩}{\mathrm{\partial }T}+kT{\mu }_1\frac{\mathrm{\partial }⟨E⟩}{\mathrm{\partial }{\mu }_1}+kT{\mu }_2\frac{\mathrm{\partial }⟨E⟩}{\mathrm{\partial }{\mu }_2}+\dots ,}$

${\displaystyle ⟨N_1^2⟩-⟨N_1{⟩}^2=kT\frac{\mathrm{\partial }⟨N_1⟩}{\mathrm{\partial }{\mu }_1}.}$

Correlations in fluctuations: The covariances of particle numbers and energy are

${\displaystyle ⟨N_1{N}_2⟩-⟨N_1⟩⟨N_2⟩=kT\frac{\mathrm{\partial }⟨N_2⟩}{\mathrm{\partial }{\mu }_1}=kT\frac{\mathrm{\partial }⟨N_1⟩}{\mathrm{\partial }{\mu }_2}.}$

${\displaystyle ⟨N_{1}E⟩-⟨N_1⟩⟨E⟩=kT\frac{\mathrm{\partial }⟨E⟩}{\mathrm{\partial }{\mu }_1},}$

Example ensembles

The usefulness of the grand canonical ensemble is illustrated in the examples below. In each case the grand potential is calculated on the basis of the relationship

${\displaystyle \mathrm{\Omega }=-kT\ln \left(\sum _{\text{microstates}}{e}^{\frac{\mu N-E}{kT}}\right)}$

which is required for the microstates' probabilities to add up to 1.

Statistics of noninteracting particles

Bosons and fermions (quantum)

In the special case of a quantum system of many non-interacting particles, the thermodynamics are simple to compute. Since the particles are non-interacting, one can compute a series of single-particle stationary states, each of which represent a separable part that can be included into the total quantum state of the system. For now let us refer to these single-particle stationary states as orbitals (to avoid confusing these "states" with the total many-body state), with the provision that each possible internal particle property (spin or polarization) counts as a separate orbital. Each orbital may be occupied by a particle (or particles), or may be empty.

Since the particles are non-interacting, we may take the viewpoint that each orbital forms a separate thermodynamic system. Thus each orbital is a grand canonical ensemble unto itself, one so simple that its statistics can be immediately derived here. Focusing on just one orbital labelled i, the total energy for a microstate of N particles in this orbital will be Nϵ_i, where ϵ_i is the characteristic energy level of that orbital. The grand potential for the orbital is given by one of two forms, depending on whether the orbital is bosonic or fermionic:

• For fermions, the Pauli exclusion principle allows only two microstates for the orbital (occupation of 0 or 1), giving a two-term series

  ${\displaystyle \begin{array}{rl}{\mathrm{\Omega }}_i& =-kT\ln (\sum _{N=0}^1{e}^{\frac{N\mu -N{ϵ}_i}{kT}})\\ & =-kT\ln (1+e^{\frac{\mu -{ϵ}_i}{kT}})\end{array}}$

• For bosons, N may be any nonnegative integer and each value of N counts as one microstate due to the indistinguishability of particles, leading to a geometric series:
  ${\displaystyle \begin{array}{rl}{\mathrm{\Omega }}_i& =-kT\ln (\sum _{N=0}^{\mathrm{\infty }}{e}^{\frac{N\mu -N{ϵ}_i}{kT}})\\ & =+kT\ln (1-e^{\frac{\mu -{ϵ}_i}{kT}}).\end{array}}$

In each case the value ${\displaystyle ⟨N_i⟩=-\frac{\mathrm{\partial }{\mathrm{\Omega }}_i}{\mathrm{\partial }\mu }}$ gives the thermodynamic average number of particles on the orbital: the Fermi–Dirac distribution for fermions, and the Bose–Einstein distribution for bosons. Considering again the entire system, the total grand potential is found by adding up the Ω_i for all orbitals.

Indistinguishable classical particles

In classical mechanics it is also possible to consider indistinguishable particles (in fact, indistinguishability is a prerequisite for defining a chemical potential in a consistent manner; all particles of a given kind must be interchangeable). We again consider placing multiple particles of the same kind into the same microstate of single-particle phase space, which we again call an "orbital". However, compared to quantum mechanics, the classical case is complicated by the fact that a microstate in classical mechanics does not refer to a single point in phase space but rather to an extended region in phase space: one microstate contains an infinite number of states, all distinct but of similar character. As a result, when multiple particles are placed into the same orbital, the overall collection of the particles (in the system phase space) does not count as one whole microstate but rather only a fraction of a microstate, because identical states (formed by permutation of identical particles) should not be overcounted. The overcounting correction factor is the factorial of the number of particles.

The statistics in this case take the form of an exponential power series

${\displaystyle \begin{array}{rl}{\mathrm{\Omega }}_{\mathrm{o}\mathrm{r}\mathrm{b}}& =-kT\ln (\sum _{N=0}^{\mathrm{\infty }}\frac{1}{N!}{e}^{\frac{N\mu -N{ϵ}_{\mathrm{o}\mathrm{r}\mathrm{b}}}{kT}})\\ & =-kT\ln (e^{e^{\frac{\mu -{ϵ}_{\mathrm{o}\mathrm{r}\mathrm{b}}}{kT}}})\\ & =-kT{e}^{\frac{\mu -{ϵ}_{\mathrm{o}\mathrm{r}\mathrm{b}}}{kT}},\end{array}}$

the value ${\displaystyle ⟨N_{\mathrm{o}\mathrm{r}\mathrm{b}}⟩=-\frac{\mathrm{\partial }{\mathrm{\Omega }}_{\mathrm{o}\mathrm{r}\mathrm{b}}}{\mathrm{\partial }\mu }}$ corresponding to Maxwell–Boltzmann statistics.

Ionization of an isolated atom

Surface ionization effect in a vaporized cesium atom at 1500 K, calculated using the method in this section (also including degeneracy). Y-axis: average number of electrons; the atom is neutral when it has 55 electrons. X-axis: energy variable, which is equal to the surface work function.

The grand canonical ensemble can be used to predict whether an atom prefers to be in a neutral state or ionized state. An atom is able to exist in ionized states with more or fewer electrons compared to neutral. As shown below, ionized states may be thermodynamically preferred depending on the environment. Consider a simplified model where the atom can be in a neutral state or in one of two ionized states (a detailed calculation also includes the degeneracy factors of the states):

• charge neutral state, with N₀ electrons and energy E₀.
• an oxidized state (N₀ − 1 electrons) with energy E₀ + ΔE_I + qϕ
• a reduced state (N₀ + 1 electrons) with energy E₀ − ΔE_A − qϕ

Here ΔE_I and ΔE_A are the atom's ionization energy and electron affinity, respectively; ϕ is the local electrostatic potential in the vacuum nearby the atom, and −q is the electron charge.

The grand potential in this case is thus determined by

${\displaystyle \begin{array}{rl}\mathrm{\Omega }& =-kT\ln (e^{\frac{\mu N_0-E_0}{kT}}+e^{\frac{\mu N_0-\mu -E_0-\mathrm{\Delta }{E}_{\mathrm{I}}-q\varphi }{kT}}+e^{\frac{\mu N_0+\mu -E_0+\mathrm{\Delta }{E}_{\mathrm{A}}+q\varphi }{kT}}).\\ & =E_0-\mu N_0-kT\ln (1+e^{\frac{-\mu -\mathrm{\Delta }{E}_{\mathrm{I}}-q\varphi }{kT}}+e^{\frac{\mu +\mathrm{\Delta }{E}_{\mathrm{A}}+q\varphi }{kT}}).\end{array}}$

The quantity −qϕ − µ is critical in this case, for determining the balance between the various states. This value is determined by the environment around the atom.

If one of these atoms is placed into a vacuum box, then −qϕ − µ = W, the work function of the box lining material. Comparing the tables of work function for various solid materials with the tables of electron affinity and ionization energy for atomic species, it is clear that many combinations would result in a neutral atom, however some specific combinations would result in the atom preferring an ionized state: e.g., a halogen atom in a ytterbium box, or a cesium atom in a tungsten box. At room temperature this situation is not stable since the atom tends to adsorb to the exposed lining of the box instead of floating freely. At high temperatures, however, the atoms are evaporated from the surface in ionic form; this spontaneous surface ionization effect has been used as a cesium ion source.

At room temperature, this example finds application in semiconductors, where the ionization of a dopant atom is well described by this ensemble. In the semiconductor, the conduction band edge ϵ_C plays the role of the vacuum energy level (replacing −qϕ), and µ is known as the Fermi level. Of course, the ionization energy and electron affinity of the dopant atom are strongly modified relative to their vacuum values. A typical donor dopant in silicon, phosphorus, has ΔE_I = 45 meV; the value of ϵ_C − µ in the intrinsic silicon is initially about 600 meV, guaranteeing the ionization of the dopant. The value of ϵ_C − µ depends strongly on electrostatics, however, so under some circumstances it is possible to de-ionize the dopant.

Meaning of chemical potential, generalized "particle number"

See also: Chemical potential

In order for a particle number to have an associated chemical potential, it must be conserved during the internal dynamics of the system, and only able to change when the system exchanges particles with an external reservoir.

If the particles can be created out of energy during the dynamics of the system, then an associated µN term must not appear in the probability expression for the grand canonical ensemble. In effect, this is the same as requiring that µ = 0 for that kind of particle. Such is the case for photons in a black cavity, whose number regularly change due to absorption and emission on the cavity walls. (On the other hand, photons in a highly reflective cavity can be conserved and caused to have a nonzero µ.)

In some cases the number of particles is not conserved and the N represents a more abstract conserved quantity:

• Chemical reactions: Chemical reactions can convert one type of molecule to another; if reactions occur then the N_i must be defined such that they do not change during the chemical reaction.
• High energy particle physics: Ordinary particles can be spawned out of pure energy, if a corresponding antiparticle is created. If this sort of process is allowed, then neither the number of particles nor antiparticles are conserved. Instead, N = (particle number - antiparticle number) is conserved. As particle energies increase, there are more possibilities to convert between particle types, and so there are fewer numbers that are truly conserved. At the very highest energies the only conserved numbers are electric charge, weak isospin, and baryon number − lepton number.

On the other hand, in some cases a single kind of particle may have multiple conserved numbers:

• Closed compartments: In a system composed of multiple compartments that share energy but do not share particles, it is possible to set the chemical potentials separately for each compartment. For example, a capacitor is composed of two isolated conductors and is charged by applying a difference in electron chemical potential.
• Slow equilibration: In some quasi-equilibrium situations it is possible to have two distinct populations of the same kind of particle in the same location, which are each equilibrated internally but not with each other. Though not strictly in equilibrium, it may be useful to name quasi-equilibrium chemical potentials which can differ among the different populations. Examples: (semiconductor physics) distinct quasi-Fermi levels (electron chemical potentials) in the conduction band and valence band; (spintronics) distinct spin-up and spin-down chemical potentials; (cryogenics) distinct parahydrogen and orthohydrogen chemical potentials.

Precise expressions for the ensemble

The precise mathematical expression for statistical ensembles has a distinct form depending on the type of mechanics under consideration (quantum or classical), as the notion of a "microstate" is considerably different. In quantum mechanics, the grand canonical ensemble affords a simple description since diagonalization provides a set of distinct microstates of a system, each with well-defined energy and particle number. The classical mechanical case is more complex as it involves not stationary states but instead an integral over canonical phase space.

Quantum mechanical

Further information on the representation of ensembles in quantum mechanics: Statistical ensemble (mathematical physics)

A statistical ensemble in quantum mechanics is represented by a density matrix, denoted by ${\displaystyle \hat{\rho }}$. The grand canonical ensemble is the density matrix

${\displaystyle \hat{\rho }=\exp (\frac{1}{kT}(\mathrm{\Omega }+{\mu }_1{\hat{N}}_1+\dots +{\mu }_s{\hat{N}}_s-\hat{H})),}$

where Ĥ is the system's total energy operator (Hamiltonian), N̂₁ is the system's total particle number operator for particles of type 1, N̂₂ is the total particle number operator for particles of type 2, and so on. exp is the matrix exponential operator. The grand potential Ω is determined by the probability normalization condition that the density matrix has a trace of one, ${\displaystyle Tr\hat{\rho }=1}$:

${\displaystyle e^{-\frac{\mathrm{\Omega }}{kT}}=\mathrm{Tr}\exp (\frac{1}{kT}({\mu }_1{\hat{N}}_1+\dots +{\mu }_s{\hat{N}}_s-\hat{H})).}$

Note that for the grand ensemble, the basis states of the operators Ĥ, N̂₁, etc. are all states with multiple particles in Fock space, and the density matrix is defined on the same basis. Since the energy and particle numbers are all separately conserved, these operators are mutually commuting.

The grand canonical ensemble can alternatively be written in a simple form using bra–ket notation, since it is possible (given the mutually commuting nature of the energy and particle number operators) to find a complete basis of simultaneous eigenstates |ψ_i⟩, indexed by i, where Ĥ|ψ_i⟩ = E_i|ψ_i⟩, N̂₁|ψ_i⟩ = N_1,i|ψ_i⟩, and so on. Given such an eigenbasis, the grand canonical ensemble is simply

${\displaystyle \hat{\rho }=\sum _i{e}^{\frac{\mathrm{\Omega }+{\mu }_1{N}_{1,i}+\dots +{\mu }_s{N}_{s,i}-E_i}{kT}}|{\psi }_i⟩⟨{\psi }_i|}$

${\displaystyle e^{-\frac{\mathrm{\Omega }}{kT}}=\sum _i{e}^{\frac{{\mu }_1{N}_{1,i}+\dots +{\mu }_s{N}_{s,i}-E_i}{kT}}.}$

where the sum is over the complete set of states with state i having E_i total energy, N_1,i particles of type 1, N_2,i particles of type 2, and so on.

Classical mechanical

Further information on the representation of ensembles in classical mechanics: Statistical ensemble (mathematical physics)

In classical mechanics, a grand ensemble is instead represented by a joint probability density function defined over multiple phase spaces of varying dimensions, ρ(N₁, … N_s, p₁, … p_n, q₁, … q_n), where the p₁, … p_n and q₁, … q_n are the canonical coordinates (generalized momenta and generalized coordinates) of the system's internal degrees of freedom. The expression for the grand canonical ensemble is somewhat more delicate than the canonical ensemble since:

• The number of particles and thus the number of coordinates n varies between the different phase spaces, and,
• it is vital to consider whether permuting similar particles counts as a distinct state or not.

In a system of particles, the number of degrees of freedom n depends on the number of particles in a way that depends on the physical situation. For example, in a three-dimensional gas of monoatoms n = 3N, however in molecular gases there will also be rotational and vibrational degrees of freedom.

The probability density function for the grand canonical ensemble is:

${\displaystyle \rho =\frac{1}{h^{n}C}{e}^{\frac{\mathrm{\Omega }+{\mu }_1{N}_1+\dots +{\mu }_s{N}_s-E}{kT}},}$

where

• E is the energy of the system, a function of the phase (N₁, … N_s, p₁, … p_n, q₁, … q_n),
• h is an arbitrary but predetermined constant with the units of energy×time, setting the extent of one microstate and providing correct dimensions to ρ.
• C is an overcounting correction factor (see below), a function of N₁, … N_s.

Again, the value of Ω is determined by demanding that ρ is a normalized probability density function:

${\displaystyle e^{-\frac{\mathrm{\Omega }}{kT}}=\sum _{N_1=0}^{\mathrm{\infty }}\dots \sum _{N_s=0}^{\mathrm{\infty }}\int \dots \int \frac{1}{h^{n}C}{e}^{\frac{{\mu }_1{N}_1+\dots +{\mu }_s{N}_s-E}{kT}}d{p}_1\dots d{q}_n}$

This integral is taken over the entire available phase space for the given numbers of particles.

Overcounting correction

Further information on phase space overcounting: Statistical ensemble (mathematical physics)

A well-known problem in the statistical mechanics of fluids (gases, liquids, plasmas) is how to treat particles that are similar or identical in nature: should they be regarded as distinguishable or not? In the system's equation of motion each particle is forever tracked as a distinguishable entity, and yet there are also valid states of the system where the positions of each particle have simply been swapped: these states are represented at different places in phase space, yet would seem to be equivalent.

If the permutations of similar particles are regarded to count as distinct states, then the factor C above is simply C = 1. From this point of view, ensembles include every permuted state as a separate microstate. Although appearing benign at first, this leads to a problem of severely non-extensive entropy in the canonical ensemble, known today as the Gibbs paradox. In the grand canonical ensemble a further logical inconsistency occurs: the number of distinguishable permutations depends not only on how many particles are in the system, but also on how many particles are in the reservoir (since the system may exchange particles with a reservoir). In this case the entropy and chemical potential are non-extensive but also badly defined, depending on a parameter (reservoir size) that should be irrelevant.

To solve these issues it is necessary that the exchange of two similar particles (within the system, or between the system and reservoir) must not be regarded as giving a distinct state of the system. In order to incorporate this fact, integrals are still carried over full phase space but the result is divided by

${\displaystyle C=N_1!N_2!\dots N_s!,}$

which is the number of different permutations possible. The division by C neatly corrects the overcounting that occurs in the integral over all phase space.

It is of course possible to include distinguishable types of particles in the grand canonical ensemble—each distinguishable type ${\displaystyle i}$ is tracked by a separate particle counter ${\displaystyle N_i}$ and chemical potential ${\displaystyle {\mu }_i}$. As a result, the only consistent way to include "fully distinguishable" particles in the grand canonical ensemble is to consider every possible distinguishable type of those particles, and to track each and every possible type with a separate particle counter and separate chemical potential.