Quasi Static Process Explanation Essay

1. Introduction

The interesting subject of quantum thermodynamics [1], which is largely based on the theory of quantum open systems [1,2], provides a theoretical framework to study quantum heat engines (QHEN). The analysis of a QHEN operating far from thermal equilibrium is an interesting and, to a great extent, an open problem. The dynamics of such systems, whose states are determined by a reduced density matrix operator out of equilibrium, is described by different approximations to the master equation [1,2,3].

As an alternative to recover thermal energy in the form of useful work on a nanoscale device, QHENs have been proposed in the literature [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]. Within the general definition of a QHEN, whose working fluid is of a quantum mechanical nature, it is important to distinguish those that have a reciprocating operation [3,19]. It has been proven [19] that under not too restrictive conditions, a reciprocating QHEN converges to a stationary limit cycle. Such a limit cycle, in direct analogy to a classical engine, can be pictured as a sequence of equilibrium states, with a trajectory determined by a minimal set of variables [1,19,20]. Interesting examples of this concept are constituted by photosynthesis in plants [21], as well as human-designed photocells [11,12], where the working substance are thermalized photons. Moreover, it has been recently proposed that if the reservoirs are also of a quantum mechanical nature, these could be prepared into quantum coherent states [11,12] or into squeezed thermal states [11,12,22]. These examples by no means constitute the only possible configurations, since a number of different designs based on alternative principles have been proposed in the literature, such as entangled states in a qubit [23] and quantum mechanical versions of the Diesel [15] and the Otto cycle [16,22,24,25]. Conceptually, a statistical ensemble of confined single-particle systems can undergo a cycle of reversible transformations driven by a generalized external field. The driving field can be a mechanical force [4,5,6,7,9] that, by tuning a confinement length-scale, modifies the inter-level spacing of the single-particle spectrum, thus inducing a sequence of transitions on the statistical population of the single-particle states, in close analogy to a classical gas confined by a piston. We have discussed generalizations of this idea in the context of relativistic Dirac particles [26]. More recently, we proposed [27] a magnetically-driven QHEN, based on the combined effects of a parabolic confining potential, representing a semiconductor quantum dot, and an external magnetic field. In the single-particle picture, this configuration possesses an exact solution in terms of effective Landau levels, which constitute a discrete spectrum, and the effective confinement length is determined by the Landau radius. Here, the inter-level distance can be modulated by tuning the external magnetic field [27], with the effect of modulating the Landau radius. An interesting variation over this idea is the magneto-strain driven QHEN that, as we recently proposed, could be implemented on an ensemble of graphene flakes [28]. In this article, we will start with a brief introduction to the theoretical framework of quantum thermodynamics. We will present a microscopic formulation of the first law of thermodynamics, both from general out-of-equilibrium considerations, as well as in the more restricted quasi-static conditions. Finally, we will present two explicit realistic examples of the application of this formalism, in the analysis of magnetically-driven quantum engines [27,28].

2. General Theory

Along this section, we will present a brief introduction to the main definitions and physical quantities involved in the theory of quantum thermodynamics. These definitions will allow us to formulate the basic theory and assumptions required for the analysis of QHENs. First, we shall introduce the reduced density matrix operator characterizing the non-equilibrium states of an open quantum system, as well as the Markovian approximation leading to the master equation that governs its time evolution. From this description, we shall derive the non-equilibrium, microscopic version of the first law of thermodynamics for open quantum systems. At last, we shall focus on the quasi-static approximation to this law, which constitutes the theoretical basis for the two explicit examples of QHENs to be discussed in Section 3 and Section 4.

In general terms, a QHEN can be pictured as a quantum open system in contact with one or more thermal reservoirs, with which the system interacts and exchanges energy and information.

As depicted schematically in Figure 1, if the open system (S) and the reservoir (B) were isolated entities, they would be described by Hamiltonian operators and , acting over states in Hilbert spaces and , respectively. Due to the interaction that mixes the degrees of freedom of both subsystems S and B, the composed Hilbert space for the combination is , with the Hamiltonian:
Let us define by the density matrix operator describing the mixed state of the combined system S and B. The reduced density matrix operator that characterizes the state of S, denoted by , is obtained by tracing over the reservoir degrees of freedom,
The equation of motion for the reduced density matrix Equation (2) is then given by:
Under general physical assumptions, mainly concerning the short correlation times for observables in the reservoir B (to be discussed later in more detail), Equation (3) can be approximated by a Markovian master equation [1,2] of the Lindblad form. Mathematically, this involves the existence of a dynamical map from the space of density matrices of the system S onto itself. This map combines unitary evolution with a dissipative term described by a super-operator and can be seen as a quantum mechanical version of the Liouville equation [1,2]:

Here, the super-operator represents the dissipative contribution arising from the interaction between the open system S and the reservoir B. Notice that in general, may include a renormalization of the Hamiltonian due to interaction between the system and the reservoir [2]. We shall come back to this point later in more detail.

The general form [1] of the super-operator representing dissipation, within the assumptions leading to the Markovian master Equation (4), can be expressed by a set of Lindblad operators , such that, for any observable :

Objectives of Thermodynamics and Heat Transfer

Arunn Narasimhan

The radiantly cooling Sun is dying to keep us as a species warm and thinking. The conduction cooling of the Earth from its center doesnt seem to affect our daily lives, but the convection winds around it do. Living amidst all this natural energy interactions trying to generate as much power as we individually can we realize we cannot beat even the horse (736 Watts). Suspecting an unfair tilt from Nature, we as a species have realized over the centuries how to harness power from nature by building tools of expediency through thought and effort. An abstract concept connects all these human and natural endeavors and conflicts.

Flip through your textbooks and you may always end up with disappointment in not actually getting to understand what exactly this concept is. Energy is the concept. And defining it exactly is the onus. But this essay is much less ambitious. It doesnt attempt an unambiguous definition for energy. It only offers a simple equation-free explanation of some principles and guidelines that teach us how to control energy interactions. Although it is carefully kept equation-free and carefree, an high school science knowledge is necessary for the reader to comprehend what follows.

Thermodynamics is a subject that talks about how one could study systems in terms of their energy interactions with its immediate neighborhood. Here a system is understood as anything that is of our interest and neighborhood is the surrounding to the system or what is not system, but which is close by to it so that it participates in the afore-said energy interactions. The energy interactions are in turn quantified (measured) using the variations in appropriate characteristics of both the system and its surroundings, called the properties. Some rules about these interactions hold true (until now) in simplifying the classification and study of these energy inventory and interactions. One of them for instance is the conservation of energy that one is taught in high schools, and later in another guise as the First Law of Thermodynamics. There is another one, termed the Second Law of Thermodynamics that talks about what is probable out of all that is possible if one were to use only the conservation laws. Of course, there are the Zeroth and the Third Laws of Thermodynamics that have an indirect bearing on these energy interactions, which we shall see later, when required.

Using these laws of thermodynamics alone one would be able to quantify or associate a number, through measurement of related properties, to these energies and their interactions. In other words, given a system, an energy inventory or balance sheet for it along with what sort of energy interaction processes or trade is possible by it shall be given by thermodynamics. However, for practical reasons if one were to ask questions like how long would a system take to reduce or increase its energy inventory from one value to another value (quantified using measurements), then thermodynamics may not give a direct answer. A thermodynamically reversible or quasi static process of energy interaction may take forever to complete - a useless result for ephemeral creatures on an energy harnessing overdrive.

Suppose in an year one has earned a periodic salary (say a dozen times) and throughout the year has spent most of it in various “expenditure” interactions, the balance in cash should tell one how much has been spent, if a proper set of addition (of all expenditure at various instances) and subtraction (of that total amount of expenditure from the total amount of income) is carried out. However, this exercise may not tell one how long each of the expenditure interaction took, although for each of this expenditure interaction one could know how much one spent. Similarly, the laws of thermodynamics are adept at telling one how much energy has been transferred in each of the energy interactions (both incoming and outgoing with respect to the system) and how much remains in the system. However, it wont give how long each of these energy interactions took to happen. For finding this one need to look closely at how such energy interactions happen.

It is our understanding that energy is transferred between systems either as heat or as work. Historically we have accepted the classification that the energy transfer that results across a finite temperature difference is heat transfer and the rest of the energy transfer interactions can be brought under one or other type of work transfer. It would seem we arent studying work transfer as a separate subject of knowledge. Owing to the acceptance of the mechanics definition of work - as equivalent to the product of the force applied and the resultant displacement of an object - and the possibility of expressing other types of energy transfers as an equivalent work transfer, we keep reading about work transfer in several other subjects throughout our academics. On the other hand, heat transfer is aggregated as a separate subject involving not only the first law but also the second law of thermodynamics and also to an extent fluid mechanics.

The answer to how long is answered at least for energy interactions of systems across finite temperature differences. The answer is found to be not a unique value in time but also depends on the mode of heat transfer. Across a finite temperature difference that could be present between a system and its surrounding, heat transfer energy interaction happens with different magnitudes depending on the mode of transfer being conduction, convection or radiation. Further, in about two centuries of concerted investigations, it has been established with enough certainty that the heat power q (in Watts) depends individually on several parameters such as the system and surrounding temperatures, their geometrical shape, their relative movement or flow, their thermo-physical properties such as specific heat, thermal conductivity, viscosity, emissivity. Using this subject knowledge of the science of heat transfer, the time taken for the heat transfer process of energy interaction can be controlled - increased or decreased - resulting in engineering. There lies the objectives of engineering heat transfer.

The objectives of heat transfer include the following:

Insulation, wherein across a finite temperature difference between the system and its surrounding, the engineer seeks to reduce the heat transfer as much as possible. In other words, a finite quantity of energy (in Joules) is made to transfer as heat very slowly in time - hence, the wattage of heat transfer is very low. By conduction mode of heat transfer, this objective is achieved for instance by using a material with very low thermal conductivity (k, W.m-1K-1) - a very poor heat conductor. The thermos flask on the other hand achieves this objective by providing an annular space of near vacuum in its walls as the resulting radiation mode of heat transfer across this space is very low.

Enhancement, wherein the converse of insulation, i.e. promotion of heat transfer is sought across a finite temperature difference. Convection mode of heat transfer is one easy way of achieving enhancement and we know of it when we blow over hot coffee to cool it. Elsewhere, big power plants use heat exchangers, devices that promote enhancement of heat transfer mainly through convection.

Temperature control, wherein the temperature of a region is required to be maintained close to a specified value, requiring both insulation and enhancement to operate at various instances of the operational sequence of a device kept in the region of interest. Cooling of electronics using phase change materials is one such example wherein temperature in the electronic components are maintained at or below their reliable operation temperature (usually about 90 degree C) by rejecting their joule heating when they are switchedon to the phase change materials and allowing the phase change material to cool (reject the heat in turn to the surroundings) when the electronic device is off. Thermoregulation of our human body through more or less blood flow to localized regions coupled with mechanisms like sweating is another example of temperature control. When this mechanism gets messed up, fever or hypothermia results.

Whether the science itself is different in finding new phenomena that is conceptually different from the existing modes of heat transfer so that they require refinement, or the engineering of devices in such scales could lead to commercially viable novel applications for the society, is the pursuit of nano-scale thermal science - an endeavor of recent prominence but fairly ancient origins.

And here we cool it, to reflect, marshal, invent, discover and rewrite…

…before the Sun is forever dead.

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