Quantum thermodynamics is the study of the relations between two independent physical theories:
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. The two independent theories address the physical phenomena of light and matter.
In 1905,
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...
argued that the requirement of consistency between
thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
and
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
leads to the conclusion that light is quantized obtaining the relation
. This paper is the dawn of
quantum
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
theory. In a few decades
quantum
In physics, a quantum (plural quanta) is the minimum amount of any physical entity (physical property) involved in an interaction. The fundamental notion that a physical property can be "quantized" is referred to as "the hypothesis of quantizati ...
theory became established with an independent set of rules. Currently quantum thermodynamics addresses the emergence of thermodynamic laws from quantum mechanics.
It differs from
quantum statistical mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is a ...
in the emphasis on dynamical processes out of equilibrium.
In addition, there is a quest for the theory to be relevant for a single individual quantum system.
Dynamical view
There is an intimate connection of quantum thermodynamics with the theory of
open quantum system In physics, an open quantum system is a quantum-mechanical system that interacts with an external quantum system, which is known as the ''environment'' or a ''bath''. In general, these interactions significantly change the dynamics of the system an ...
s.
Quantum mechanics inserts dynamics into thermodynamics, giving a sound foundation to finite-time-thermodynamics.
The main assumption is that the entire world is a large closed system, and therefore, time evolution
is governed by a unitary transformation generated by a global
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
. For the combined system
bath scenario, the global Hamiltonian can be decomposed into:
:
where
is the system Hamiltonian,
is the bath Hamiltonian and
is the system-bath interaction.
The state of the system is obtained from a partial trace over the combined system and bath:
.
Reduced dynamics is an equivalent description of the system dynamics utilizing only system operators.
Assuming
Markov property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov propert ...
for the dynamics the basic equation of motion for an open quantum system is the
Lindblad equation
In quantum mechanics, the Gorini–Kossakowski–Sudarshan–Lindblad equation (GKSL equation, named after Vittorio Gorini, Andrzej Kossakowski, George Sudarshan and Göran Lindblad), master equation in Lindblad form, quantum Liouvillian, or Li ...
(GKLS):
:
is a (
Hermitian {{Short description, none
Numerous things are named after the French mathematician Charles Hermite (1822–1901):
Hermite
* Cubic Hermite spline, a type of third-degree spline
* Gauss–Hermite quadrature, an extension of Gaussian quadrature m ...
)
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
part and
:
:
is the dissipative part describing implicitly through system operators
the influence of the bath on the system.
The
Markov property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov propert ...
imposes
that the system and bath are uncorrelated at all times
.
The L-GKS equation is unidirectional and leads any initial state
to a steady state solution which is an invariant of the equation of motion
.
The
Heisenberg picture
In physics, the Heisenberg picture (also called the Heisenberg representation) is a formulation (largely due to Werner Heisenberg in 1925) of quantum mechanics in which the operators (observables and others) incorporate a dependency on time, bu ...
supplies a direct link to quantum thermodynamic observables. The dynamics of a system observable represented by the operator,
, has the form:
:
where the possibility that the operator,
is explicitly time-dependent, is included.
Emergence of time derivative of first law of thermodynamics
When
the
first law of thermodynamics
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amoun ...
emerges:
:
where power is interpreted as
and the heat current
.
Additional conditions have to be imposed on the dissipator
to be consistent with thermodynamics.
First the invariant
should become an equilibrium
Gibbs state In probability theory and statistical mechanics, a Gibbs state is an equilibrium probability distribution which remains invariant under future evolution of the system. For example, a stationary or steady-state distribution of a Markov chain, such a ...
.
This implies that the dissipator
should commute with the unitary part generated by
.
In addition an equilibrium state is stationary and stable. This assumption is used to derive
the Kubo-Martin-Schwinger stability criterion for thermal equilibrium i.e.
KMS state
In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo–Martin– Schwinger state or, more commonly, a KMS s ...
.
A unique and consistent approach is obtained by deriving the generator,
, in the weak system bath
coupling limit.
In this limit, the interaction energy can be neglected. This approach represents a thermodynamic idealization: it allows energy transfer, while keeping a tensor product separation
between the system and bath, i.e., a quantum version of an
isothermal
In thermodynamics, an isothermal process is a type of thermodynamic process in which the temperature ''T'' of a system remains constant: Δ''T'' = 0. This typically occurs when a system is in contact with an outside thermal reservoir, and a ...
partition.
Markovian behavior involves a rather complicated cooperation between system and bath dynamics. This means that in
phenomenological treatments, one cannot combine arbitrary system Hamiltonians,
, with a given L-GKS generator. This observation is particularly important in the context of quantum thermodynamics,
where it is tempting to study Markovian dynamics with an arbitrary control Hamiltonian. Erroneous derivations of the quantum master equation can easily lead to a violation of the laws of thermodynamics.
An external perturbation modifying the Hamiltonian of the system will also modify the heat flow. As a result, the L-GKS generator has to be renormalized. For a slow change, one can adopt the adiabatic approach and use the instantaneous system’s Hamiltonian to derive
. An important class of problems in quantum thermodynamics is periodically driven systems. Periodic
quantum heat engines A quantum heat engine is a device that generates power from the heat flow between hot and cold reservoirs.
The operation mechanism of the engine can be described by the laws of quantum mechanics.
The first realization of a quantum heat engine was ...
and power-driven
refrigerators
A refrigerator, colloquially fridge, is a commercial and home appliance consisting of a thermally insulated compartment and a heat pump (mechanical, electronic or chemical) that transfers heat from its inside to its external environment so th ...
fall into this class.
A reexamination of the time-dependent heat current expression using quantum transport techniques has been proposed.
A derivation of consistent dynamics beyond the weak coupling limit has been suggested.
Phenomenological formulations of irreversible quantum dynamics consistent with the second law and implementing the geometric idea of "steepest entropy ascent" or "gradient flow" have been suggested to model relaxation and strong coupling.
Emergence of the second law
The
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
is a statement on the irreversibility of dynamics or, the breakup of time reversal symmetry (
T-symmetry
T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal,
: T: t \mapsto -t.
Since the second law of thermodynamics states that entropy increases as time flows toward the future ...
). This should be consistent with the empirical direct definition: heat will flow spontaneously from a hot source to a cold sink.
From a static viewpoint, for a closed quantum system, the 2nd law of thermodynamics is a consequence of the unitary evolution. In this approach, one accounts for the entropy change before and after a change in the entire system. A dynamical viewpoint is based on local accounting for the
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
changes
in the subsystems and the entropy generated in the baths.
Entropy
In thermodynamics,
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
is related to the amount of energy of a system that can be converted into mechanical work in a concrete process.
In quantum mechanics, this translates to the ability to measure and manipulate the system based on the information gathered by measurement. An example is the case of
Maxwell’s demon
Maxwell's demon is a thought experiment that would hypothetically violate the second law of thermodynamics. It was proposed by the physicist James Clerk Maxwell in 1867. In his first letter Maxwell called the demon a "finite being", while the ' ...
, which has been resolved
by
Leó Szilárd
Leo Szilard (; hu, Szilárd Leó, pronounced ; born Leó Spitz; February 11, 1898 – May 30, 1964) was a Hungarian-German-American physicist and inventor. He conceived the nuclear chain reaction in 1933, patented the idea of a nuclear ...
.
The
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
of an observable is associated with the complete projective measurement of an observable,
, where the operator
has a spectral decomposition:
where
is the projection operators of the eigenvalue
.
The probability of outcome j is
The entropy associated with the observable
is the
Shannon entropy
Shannon may refer to:
People
* Shannon (given name)
* Shannon (surname)
* Shannon (American singer), stage name of singer Shannon Brenda Greene (born 1958)
* Shannon (South Korean singer), British-South Korean singer and actress Shannon Arrum W ...
with respect to the possible outcomes:
:
The most significant observable in thermodynamics is the energy represented by the Hamiltonian operator
, and its associated energy entropy,
.
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
suggested to single out the most informative observable to characterize the entropy of the system. This invariant is obtained by minimizing the entropy with respect to all possible observables. The most informative observable operator commutes with the state of the system. The
entropy of this observable is termed the
Von Neumann entropy
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ...
and is equal to:
:
As a consequence,
for all observables. At thermal equilibrium the energy
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
is equal to the
von Neumann entropy
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ...
:
.
is invariant to a unitary transformation changing the state. The
Von Neumann entropy
In physics, the von Neumann entropy, named after John von Neumann, is an extension of the concept of Gibbs entropy from classical statistical mechanics to quantum statistical mechanics. For a quantum-mechanical system described by a density matrix ...
is additive only for a system state that is composed of a tensor product
of its subsystems:
:
Clausius version of the II-law
No process is possible whose sole result is the transfer of heat from a body of lower temperature to a body of higher temperature.
This statement for N-coupled heat baths in steady state becomes:
:
A dynamical version of the II-law can be proven, based on
Spohn’s inequality
:
which is valid for any L-GKS generator, with a stationary state,
.
Consistency with thermodynamics can be employed to verify quantum dynamical models of transport. For example, local models for networks where local L-GKS equations are connected through weak links have been shown to violate the
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
.
Quantum and thermodynamic adiabatic conditions and quantum friction
Thermodynamic
adiabatic process
In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
es have no entropy change. Typically, an external control modifies
the state. A quantum version of an
adiabatic process
In thermodynamics, an adiabatic process (Greek: ''adiábatos'', "impassable") is a type of thermodynamic process that occurs without transferring heat or mass between the thermodynamic system and its environment. Unlike an isothermal process, ...
can be modeled by an externally controlled time dependent
Hamiltonian
. If the system is isolated, the dynamics are unitary, and therefore,
is
a constant. A quantum adiabatic process is defined by the energy entropy
being constant.
The quantum
adiabatic
condition is therefore equivalent to no net change in the population of the instantaneous energy levels.
This implies that the Hamiltonian should commute with itself at different times:
.
When the adiabatic conditions are not fulfilled, additional work is required to reach the final control
value. For an isolated system, this work is recoverable, since the dynamics is unitary and can be reversed. In this case, quantum friction can be suppressed using
shortcuts to adiabaticity
Shortcuts to adiabaticity (STA) are fast control protocols to drive the dynamics of system without relying on the adiabatic theorem. The concept of STA was introduced in a 2010 paper by Xi Chen et al.
Their design can be achieved using a variety ...
as demonstrated in the laboratory using a unitary Fermi gas in a time-dependent trap.
The
coherence stored in the off-diagonal elements of the density operator carry the required information
to recover the extra energy cost and reverse the dynamics. Typically, this energy is not recoverable, due
to interaction with a bath that causes energy dephasing. The bath, in this case, acts like a measuring
apparatus of energy. This lost energy is the quantum version of friction.
Emergence of the dynamical version of the third law of thermodynamics
There are seemingly two independent formulations of the
third law of thermodynamics
The third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: This constant value cannot depend on any other parameters characterizing the closed system, such as pressure or applied magnetic fiel ...
both originally were stated by
Walther Nernst
Walther Hermann Nernst (; 25 June 1864 – 18 November 1941) was a German chemist known for his work in thermodynamics, physical chemistry, electrochemistry, and solid state physics. His formulation of the Nernst heat theorem helped pave the wa ...
. The first formulation is known as the
Nernst heat theorem
The Nernst heat theorem was formulated by Walther Nernst early in the twentieth century and was used in the development of the third law of thermodynamics.
The theorem
The Nernst heat theorem says that as absolute zero is approached, the entropy ...
, and can be phrased as:
*The entropy of any pure substance in thermodynamic equilibrium approaches zero as the temperature approaches zero.
The second formulation is dynamical, known as the ''unattainability principle''
*It is impossible by any procedure, no matter how idealized, to reduce any assembly to
absolute zero
Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...
temperature in a finite number of operations.
At steady state the
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
implies that the total
entropy production
Entropy production (or generation) is the amount of entropy which is produced in any irreversible processes such as heat and mass transfer processes including motion of bodies, heat exchange, fluid flow, substances expanding or mixing, anelastic d ...
is non-negative.
When the cold bath approaches the absolute zero temperature,
it is necessary to eliminate the
entropy production
Entropy production (or generation) is the amount of entropy which is produced in any irreversible processes such as heat and mass transfer processes including motion of bodies, heat exchange, fluid flow, substances expanding or mixing, anelastic d ...
divergence at the cold side
when
, therefore
::
For
the fulfillment of the
second law depends on the
entropy production
Entropy production (or generation) is the amount of entropy which is produced in any irreversible processes such as heat and mass transfer processes including motion of bodies, heat exchange, fluid flow, substances expanding or mixing, anelastic d ...
of the other baths,
which should compensate for the negative
entropy production
Entropy production (or generation) is the amount of entropy which is produced in any irreversible processes such as heat and mass transfer processes including motion of bodies, heat exchange, fluid flow, substances expanding or mixing, anelastic d ...
of the cold bath.
The first formulation of the third law modifies this restriction.
Instead of
the third law imposes
,
guaranteeing that at absolute zero the entropy production at the cold bath is zero:
.
This requirement leads to the scaling condition of the heat current
.
The second formulation, known as the unattainability principle can be rephrased as;
*No refrigerator can cool a system to
absolute zero
Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibration ...
temperature at finite time.
The dynamics of the cooling process is governed by the equation:
::
where
is the heat capacity of the bath. Taking
and
with
, we can quantify this formulation by evaluating the characteristic exponent
of the cooling process,
::
This equation introduces the relation between the characteristic exponents
and
. When
then the bath is cooled to zero temperature in a finite time, which implies a valuation of the third law. It is apparent from the last equation, that the unattainability principle is more restrictive than the
Nernst heat theorem
The Nernst heat theorem was formulated by Walther Nernst early in the twentieth century and was used in the development of the third law of thermodynamics.
The theorem
The Nernst heat theorem says that as absolute zero is approached, the entropy ...
.
Typicality as a source of emergence of thermodynamic phenomena
The basic idea of quantum typicality is that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time.
This is meant to apply to Schrödinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average.
Quantum ergodic theorem originated by
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
is a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of termed normal typicality, i.e. the statement that, for typical large systems, every initial wave function
from an energy shell is ‘normal’: it evolves in such a way that
for most t, is macroscopically equivalent to the micro-canonical density matrix.
Resource theory
The
second law of thermodynamics
The second law of thermodynamics is a physical law based on universal experience concerning heat and Energy transformation, energy interconversions. One simple statement of the law is that heat always moves from hotter objects to colder objects ( ...
can be interpreted as quantifying state transformations which are statistically unlikely so that they become effectively forbidden. The second law typically applies to systems composed of many particles interacting; Quantum thermodynamics resource theory is a formulation of thermodynamics in the regime where it can be applied to a small number of particles interacting with a heat bath. For processes which are cyclic or very close to cyclic, the second law for microscopic systems takes on a very different form than it does at the macroscopic scale, imposing not just one constraint on what state transformations are possible, but an entire family of constraints. These second laws are not only relevant for small systems, but also apply to individual macroscopic systems interacting via long-range interactions, which only satisfy the ordinary second law on average. By making precise the definition of thermal operations, the laws of thermodynamics take on a form with the first law defining the class of thermal operations, the zeroth law emerging as a unique condition ensuring the theory is nontrivial, and the remaining laws being a monotonicity property of generalised free energies.
Engineered reservoirs
Nanoscale allows for the preparation of quantum systems in physical states without classical analogs. There, complex out-of-equilibrium scenarios may be produced by the initial preparation of either the working substance or the reservoirs of quantum particles, the latter dubbed as "engineered reservoirs". There are different forms of engineered reservoirs. Some of them involve subtle quantum coherence or correlation effects,
[M. O. Scully, M. Suhail Zubairy, G. S. Agarwal1, H. Walther, Extracting Work from a Single Heat Bath via Vanishing Quantum Coherence, Science 299, 862-864 (2003).](_blank)
/ref>[G. Manzano, F. Galve, R. Zambrini, and J. M. R. Parrondo, Entropy production and thermodynamic power of the squeezed thermal reservoir, Phys. Rev. E 93, 052120 (2016).](_blank)
/ref>[R. J. de Assis, T. M. de Mendonça, C. J. Villas-Boas, A. M. de Souza, R. S. Sarthour, I. S. Oliveira, and N. G. de Almeida, Efficiency of a Quantum Otto Heat Engine Operating under a Reservoir at Effective Negative Temperatures, Phys. Rev. Lett. 122, 240602 (2019).](_blank)
/ref> while others rely solely on nonthermal classical probability distribution functions.[H. Pothier, S. Guéron, Norman O. Birge, D. Esteve, and M. H. Devoret, Energy Distribution Function of Quasiparticles in Mesoscopic Wires, Phys Rev. Lett. 79, 3490 (1997).](_blank)
/ref>
[Y.-F. Chen, T. Dirks, G. Al-Zoubi, N. O. Birge, and N. Mason, Nonequilibrium Tunneling Spectroscopy in Carbon Nanotubes, Phys. Rev. Lett. 102, 036804 (2009).](_blank)
/ref>
[C. Altimiras, H. le Sueur, U. Gennser, A. Cavanna, D. Mailly, and F. Pierre, Tuning Energy Relaxation along Quantum Hall Channels, Phys. Rev. Lett. 105, 226804 (2010).](_blank)
/ref>
[N. Bronn and N. Maso, Spatial dependence of electron interactions in carbon nanotubes, Phys. Rev. B 88, 161409(R) (2013).](_blank)
/ref>
The latter are dubbed nonequilibrium incoherent reservoirs.[S. E. Deghi and R. A. Bustos-Marún, Entropy current and efficiency of quantum machines driven by nonequilibrium incoherent reservoirs, Phys. Rev. B 102, 045415 (2020).](_blank)
/ref> Interesting phenomena may emerge from the use of engineered reservoirs such as efficiencies greater than the Otto limit,
violations of Clausius inequalities,[R. Sánchez , J. Splettstoesser, and R. S. Whitney, Nonequilibrium System as a Demon, Phys. Rev. Lett. 123, 216801 (2019).](_blank)
/ref> or simultaneous extraction of heat and work from the reservoirs.
In general, the thermodynamics and efficiency of such systems require particular analysis. However, for the special case of NIR, the efficiency of steady-state quantum machines connected to them can be treated within a unified picture.
See also
* Quantum statistical mechanics
Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is a ...
* Thermal quantum field theory
In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature.
...
References
Further reading
Deffner, Sebastian and Campbell, Steve. "Quantum Thermodynamics: An introduction to the thermodynamics of quantum information", (Morgan & Claypool Publishers, 2019).
F. Binder, L. A. Correa, C. Gogolin, J. Anders, G. Adesso (eds.) "Thermodynamics in the Quantum Regime. Fundamental Aspects and New Directions." (Springer 2018)
Jochen Gemmer, M. Michel, and Günter Mahler. "Quantum thermodynamics. Emergence of thermodynamic behavior within composite quantum systems. 2." (2009).
Petruccione, Francesco, and Heinz-Peter Breuer. The theory of open quantum systems. Oxford university press, 2002.
External links
*Go to "
Concerning an Heuristic Point of View Toward the Emission and Transformation of Light
'" to read an English translation of Einstein's 1905 paper. (Retrieved: 2014 Apr 11)
{{Branches of physics
Quantum mechanics
Thermodynamics
Non-equilibrium thermodynamics
Philosophy of thermal and statistical physics