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Irreversible Thermodynamics

Since its early stages as a science thermodynamics provides bounds and limits for processes exchanging work and heat. Usually these limits are obtained from the study of reversible processes. Irreversibilities due to finite time or finite process rates are neglected and processes are effectively treated as if they would proceed without loss and at infinite slow speed. However, this contradicts the dynamics of the world we are living in. Although the performance limits of reversible processes provide upper bounds for real, irreversible processes, they may not be guides useful enough for the evaluation of real processes. Existing heat engines, for example, seldom attain more than a fraction of the reversible Carnot efficiency. A number of approaches - Finite-Time Thermodynamics and Endoreversible Thermodynamics - have been developed to overcome that shortcoming and to provide better limits and bounds.

Are there theoretical bounds for the performance of thermodynamic processes under the constraints of finite process times or rates?

If such bounds exist, what are the optimal operating conditions necessary to reach these bounds?

This kind of questions is indeed important for the future development of technologies and has thus inspired researchers to conduct a wide range of scientific inquiries leading to new non-equilibrium theories in thermodynamics. A variety of novel theoretical techniques have been developed to model the performance of for instance automobile engines, refrigerators, heat pumps, chemical processes and solar cells. The aim of these investigations is to identify the main thermodynamic features of the system in order to make a model as simple as possible. The idea is to reduce the mathematical description and computational effort and yet to find more realistic optima and bounds for the operation of a thermodynamical system. The concept of `endoreversibility' has proven to be a powerful tool for the construction of models with the desired qualities. Endoreversible systems basically are composed of internally reversible subsystems with (irreversible) interactions between them. The losses due to the finite times or rates of processes are located in the interactions alone. The hypothesis of endoreversibility simplifies the expenditure for the analysis essentially. Nontheless a general theory describing the non-equilibrium flows of energy, matter and other extensities between systems and its accompanying entropy production remains an open challenge.

Recent Publications

Optimal control in a quantum cooling problem
Salamon, Peter and Hoffmann, Karl Heinz and Tsirlin, Anatoly
Applied Mathematics Letters (in press) (2012)

The optimal control for cooling a quantum harmonic oscillator by controlling its frequency is considered. It is shown that this singular problem may be transformed with the proper choice of coordinates to an equivalent problem which is no longer singular. The coordinates used are sufficiently simple that a graphical solution is possible and eliminates the need to use a Weierstrass-like approach to show optimality. The optimal control of this problem is of significance in connection with cooling physical systems to low temperatures. It is also mathematically significant in showing the power and limitations of coordinate transformations for attacking apparently singular problems.

Change of state variables in the problems of parametrc control of oscillators
Tsirlin, Anatoly M. and Salamon, Peter and Hoffmann, Karl Heinz
Avtomatika i Telemekhanika (8): 53--64 (2011)

Решены задачи оптимального параметрического управления одиночным осциллятором и ансамблем квантовых осцилляторов. На их примере продемонстрированы возможности метода перехода к новым переменным пространства состояний управляемой системы.

Change of State Variables in the Problems of Parametric Control of Oscillators
Tsirlin, Anatoly M. and Salamon, Peter and Hoffmann, Karl Heinz
Automation and Remote Control 72(8): 1627--1638 (2011) ; ISSN:0005-1179

The problems of optimal parametric control of a single oscillator and an assembly of quantum oscillators were solved and used by way of example to demonstrate the potentialities of the method of transition to the new variables of the state space of the controlled system.

Time-optimal controls for frictionless cooling in harmonic traps
Hoffmann, Karl Heinz and Salamon, Peter and Rezek, Yair and Kosloff, Ronnie
Europhysics Letters 96(6): 60015-1--6 (2011)

Fast adiabatic cooling procedures have important implications for the attainability of absolute zero. While traditionally adiabatically cooling a system is associated with slow thermal processes, for the parametric quantum harmonic oscillator fast frictionless processes are known, which transfer a system from an initial thermal equilibrium at one temperature into thermal equilibrium at another temperature. This makes such systems special tools in analyzing the bounds on fast cooling procedures. Previous discussions of those systems used frictionless cooling assuming real frequencies of the oscillator. Using a control with imaginary frequencies (repulsive potential) revises previous implications for the possible operation of a quantum refrigerator. Here we discuss these requisite revisions in the context of the third law of thermodynamics. In addition to minimum time controls, which are always of the bang-bang form, fast frictionless processes with a continuous variation of the frequency have been presented previously in the literature. Such continuous variation controls have been experimentally verified by cooling a Bose-Einstein condensate, while minimum time controls still await verification. As some implementations may indeed not be able to implement the instantaneous jumps in frequency required by bang-bang controls, constraining the rate of change in the frequency calls for ramped bang-bang solutions. We present such solutions and compare their performance to the continuous controls used in the experiment.

Accuracy of coarse grained Markovian dynamics
Hoffmann, Karl Heinz and Salamon, Peter
Physica A: Statistical Mechanics and its Applications 390: 3086--3094 (2011)

Markov chain models on a mesoscopic level are a widely used description for complex systems. They are based on the assumption that certain sets of microstates can be coarse grained as their internal dynamics is faster than the time scales considered in the modeling. Here we analyze quantitatively the errors made by using lumping techniques and present the first rigorous proof for bounds on such errors. Our bounds express the deviations from a full microscopic description for all subsequent time steps in terms of the deviations in the first time step.

Maximum work in minimum time from a conservative quantum system
Salamon, P. and Hoffmann, K. H. and Rezek, Y. and Kosloff, R.
Physical Chemistry Chemical Physics11: 1027--1032 (2009) ; DOI: 10.1039/b816102j

This paper considers the problem of obtaining maximum work from a conservative quantum system corresponding to a given change in an external parameter in the Hamiltonian. The example we present is a non-interacting collection of harmonic oscillators with a shared frequency o which changes from a given initial to a given final value. The example is interesting for its role in experiments at ultra-low temperatures and for probing finite-time versions of the third law of thermodynamics. It is also the simplest system displaying quantum friction, which represents loss mechanisms in any reversible prelude to a thermal process. The example leads to a new type of availability. It is also the first example of a minimum time for transitions between thermal states of a thermodynamic system.

The quantum refrigerator: The quest for absolute zero
Rezek, Y. and Salamon, P. and Hoffmann, K. H. and Kosloff, R.
Europhysics Letters 85: 1--5 (2009)

The emergence of the laws of thermodynamics from the laws of quantum mechanics is an unresolved issue. The generation of the third law of thermodynamics from quantum dynamics is analysed. The scaling of the optimal cooling power of a reciprocating quantum refrigerator is sought as a function of the cold bath temperature as $T_C ightarrow 0$. The working medium consists of noninteracting particles in a harmonic potential. Two closed-form solutions of the refrigeration cycle are analyzed, and compared to a numerical optimization scheme, focusing on cooling toward zero temperature. The optimal cycle is characterized by linear relations between the heat extracted from the cold bath, the energy level spacing of the working medium and the temperature. The scaling of the optimal cooling rate is found to be proportional to $T_C^{3/2}$ giving a dynamical interpretation to the third law of thermodynamics.

A graphic based interface to Endoreversible Thermodynamics
Wagner, Katharina
Master Thesis, TU Chemnitz, 2008

The object of this thesis is a graphic based interface to endoreversible thermodynamics. It is meant to enable the user to visually create endoreversible systems and add the properties of the system by choosing features from a list and in form of equations. Then an equation system is built and the power output and efficiency of the endoreversible system is calculated and plotted. To illustrate the functions of the interface, some examples of heat and chemical engines are discussed.

Endoreversible Thermodynamics: A Tool for Simulating and Comparing Processes of Discrete Systems
W. Muschik and K. H. Hoffmann
Journal of Non-Equilibrium Thermodynamics 31(3): 293--317 (2006) ; ISSN: 0340-0204

Endoreversible thermodynamics is concerned with reversible sub-systems which are in irreversible interaction with each other. Consequently, endoreversible thermodynamics represents the analogue for discrete systems to the local equilibrium hypothesis in continuum thermodynamics. Here a real cyclic 2-reservoir process is simulated by endoreversible model processes. Simulation means, that the simulating process has the same net heat exchanges, cycle time, power, entropy production, and efficiency as the original one. By introducing process-independent heat conduction coefficients as a constraint for the irreversible interaction, a family of comparative endoreversible processes is generated including the simulation of the original process. This procedure allows to compare process parameters of the family of comparative processes to those of the original one. The fraction ``power of the real process over the maximal power inbetween the comparative family'' is introduced as a parameter describing the process excellence.

Simulation diskreter Markov-Prozesse zweiter Stufe
Bauer, M.
Bachelor Thesis, 2006

Simulationen von Markov-Prozessen erster Stufe sind in vielen Anwendungsbereichen bekannt, Erkenntnisse über die Eigenschaften von Prozessen zweiter Markov-Stufe sind jedoch rar. Wichtige Merkmale sollen durch die Simulation diskreter Markov-Prozesse zweiter Stufe herausgearbeitet werden. An erster Stelle steht dabei die Entwicklung eines einfachen Modells zur Simulation von Markov-Prozessen zweiter Stufe mittels eines Random Walkers, der über ein Gedächtnis verfügt. Das Modell bildet dabei zeit- und ortsdiskrete Prozesse ab und hält die Möglich- keit zur Simulation äuß{}erer Kräfte offen. Im Weiteren geben Untersuchungen der zeitlichen Entwicklung der Ortsverteilung und des Entropieverlaufs an so simulierten Zufallswanderern Aufschluss über Ähnlichkeiten und Unterschiede zu Markov-Prozessen erster Stufe. Dabei werden analytische Abschätzun- gen zur Bestätigung der simulierten Daten genutzt und das Langzeitverhalten der Prozesse untersucht und mit dem von Markov-Prozessen erster Stufe verglichen. Weiterhin wird die Abhängigkeit der Stärke des Gedächtnisses auf das Verhalten der Zufallswanderer genauer betrachtet.