We start in this chapter arguing why quantum probability is a good candidate for modelling purposes in decision-making contexts. The quantum formalism, in this chapter, centres around the argument that such formalism can accommodate paradoxical outcomes in decision making. Quantum probability offers a response to those decision-making contexts where a consistent violation of the law of total probability occurs. Strong results have been obtained in decision-making applications and we go into some detail to discuss the so-called QQ equality and the Aumann theorem.

One of the main purposes of this chapter is to explain, albeit in an abstract manner, how quantum physics–like models of the economics-finance contexts would differ from quantum math-like (or simply, quantum-like) models. For this, the chapter begins by considering, what may be called, the “physical” foundations of quantum theory. These include the foundations pertaining to the theoretical, experimental, and interpretational aspects of quantum theory. With reference to the physical foundations, the chapter elaborates on certain expectations from agent-centric economics-finance models to qualify as “quantum physics–analogous”. Then, by briefly reviewing some of the prominent theories of analogical arguments and reasoning from the philosophy of science (for instance, Aristotle’s theory, Hesse’s theory, Gentner’s structure-mapping theory and Bartha’s articulation model), the chapter ends by proposing a strategy for the systematic construction of quantum physics–analogous models of economics and finance.

In this last chapter of the book, we keep coming back to the potential function and we attempt to connect it to more precise ideas in finance, including that of the agent heterogeneities. We also initiate a discussion on agent behaviour and causality and nonlocality. Our last words in this book will be centred on what comes next. One of the key queries we have is whether we can consider more complicated real potentials in the two-slit interference experiment with agents (and the agent two-preference interference). The other one is centred around the investigation on the nonexistence of “spooky” free will of the individual agents.

This chapter attempts to expound on basic and essential ideas (for further use in the book) from both classical and quantum mechanics. The chapter is somewhat technical in nature but only requires an elemental knowledge of calculus. The first three sections take a review of some of the elements of classical mechanics and classical statistical mechanics – the Euler–Lagrange and the Hamilton–Jacobi equations, the idea of an ensemble in the classical context, and the continuity equation for particle density. The remaining part is devoted to the elements of quantum mechanics – the connection between the Hamilton–Jacobi equation and the Schrödinger equation, the idea of an ensemble in the quantum context, the free particle wave function and operators, the uncertainty principle and the idea of the expectation value of an operator, and the concept of a wave packet.

This chapter (together with the next one) introduces probably the highlight of the book, i.e. it attempts to answer the important question: what can we now do with the quantum-physics like stance? An immediate, almost obvious, discussion centres around the analogies with the famed double-slit experiment. We set ourselves the task of answering how we can begin to enumerate, quite precisely, analogies between electrons and agents. As the reader will find out, we will need to move over several (important) hurdles, one of them being the perennially difficult analogy we need to make with the Planck constant. We then proceed in shaping the idea of two-preference interference, a concept of paramount importance in our quest to properly define the quantum physics–like research direction.

This chapter provides for a summary overview of some of the great movements in economic science. We discuss theory falsification and the historical role of the observable in economics. We provide for a brief overview of behavioural and experimental economics, as well as computational and neuroeconomics. We conclude the chapter with some ideas on the value of information in the price process.

In this chapter we provide simulations of Bohmian trajectories and probability distributions for a variety of linear potentials. We also discuss the possible physics-analogous interpretations of the model parameters from the previous chapter, in terms of which these simulations are shown. We hope this chapter can be inspirational to all those readers who also believe that the quantum physics– like research orientation is really the next important step to take.

This chapter discusses important notions which aid in formalizing how prices move toward equilibrium. We tackle the concepts of price discovery and price formation. The Kyle and Glosten and Milgrom models are discussed. We also go into some detail on the Rosenow model.

This chapter starts with the idea of a quantized version of supply and demand. We also provide for a discussion why quantum probability is useful in both option pricing and investment choice decisions. We also discuss briefly quantum-like Bayesian networks within the setting of missing financial data. We also hint toward the usefulness of open systems in social science and then dive into the very important observation that Black–Scholes Hamiltonians are non-Hermitian. Toward the end of the chapter, we give an overview of models (with and without memory and (a)symmetry between past and future times). We give some examples and then conclude the chapter with thoughts on the limitations of mathematical modelling.

learning Outcomes

After reading this chapter, the reader will be able to

• Understand the meaning of three processes of heat flow: conduction, convection, and radiation

• Know about thermal conductivity, diffusivity, and steady-state condition of a thermal conductor

• Derive Fourier's one-dimensional heat flow equation and solve it in the steady state

• Derive the mathematical expression for the temperature distribution in a lagged bar

• Derive the amount of heat flow in a cylindrical and a spherical thermal conductor

• Solve numerical problems and multiple choice questions on the process of conduction of heat

6.1 Introduction

Heat is the thermal energy transferred between different substances that are maintained at different temperatures. This energy is always transferred from the hotter object (which is maintained at a higher temperature) to the colder one (which is maintained at a lower temperature). Heat is the energy arising due to the movement of atoms and molecules that are continuously moving around, hitting each other and other objects. This motion is faster for the molecules with a largeramount of energy than the molecules with a smaller amount of energy that causes the former to have more heat. Transfer of heat continues until both objects attain the same temperature or the same speed. This transfer of heat depends upon the nature of the material property determined by a parameter known as thermal conductivity or coefficient of thermal conduction. This parameter helps us to understand the concept of transfer of thermal energy from a hotter to a colder body, to differentiate various objects in terms of the thermal property, and to determine the amount of heat conducted from the hotter to the colder region of an object. The transfer of thermal energy occurs in several situations:

• When there exists a difference in temperature between an object and its surroundings,

• When there exists a difference in temperature between two objects in contact with each other, and

• When there exists a temperature gradient within the same object.

Learning Outcomes

After reading this chapter, the reader will be able to

• Express the meaning of sphere of influence and collision frequency

• Derive the distribution function for the free paths among the molecules and demonstrate the concept of mean free path

• Calculate the expression for mean free path following Clausius and Maxwell

• Derive the expression for pressure exerted by a gas using the survival equation

• Calculate the expressions for viscosity, thermal conductivity, and diffusion coefficient of a gaseous system

• Demonstrate Brownian motion with its characteristics and calculate the mean square displacement of a particle executing Brownian motion

• State the idea of a random walk problem

• Solve numerical problems and multiple choice questions on the mean free path, viscosity, thermal conduction, diffusion, Brownian motion, and random walk

4.1 Introduction

The molecules of an ideal gas are considered as randomly moving point particles. From the concept of kinetic theory of gases (KTG), it is well established that even at room temperature, such point molecules of the ideal gas move at very large speeds. The average value of this speed can be determined assuming that the molecules obey Maxwell's speed distribution law and is given by the following expression