What is Computational Physics?

As the name suggests, computational physics is a complex intermingling of physical, mathematical, and computational principles. The physical world around us, from the interaction between atoms and molecules to the majestic dance of galaxies millions of light years away from the Earth, can be modeled mathematically based on fundamental physical principles. These mathematical equations are usually complex and are hard to solve analytically to obtain solutions. Computational physics employs applied mathematics and computational algorithms to solve these partial differential equations. Solving any computational physics problem consists of preparing a mathematical model for the problem in hand based on the physical principles involved. These mathematical models are then solved numerically using numerical algorithms designed and implemented inside a computer program. The field of computational physics is quite diverse. It has several applications, including fluid dynamics, astrophysics, molecular dynamics, environmental modeling, nuclear cleanup, the design of materials, ground-water transport, the nature of elementary particles, medical imaging, and energy management.

The Role of Physics

Over millennia of experiences and perception of physical reality, humanity has attempted to formulate the "Laws of Nature," also known as the fundamental laws of physics. Through empirical exploration and theoretical analysis, humans have sought to articulate and comprehend the elementary patterns and behaviors governing the universe.

The journey to formulate these laws began with early civilizations documenting observations of celestial movements, the interaction between different forces, and their impact on matter. From the ancient Greeks to the Renaissance thinkers and beyond, each era contributed to a growing understanding of the underlying rules that govern the universe.

As science has advanced, our ability to explore the mysteries of the universe has also improved. Newton's laws of motion, Maxwell's equations of electromagnetism, Einstein's theory of relativity, and quantum mechanics all represent monumental milestones in this quest for understanding the universe.

Mathematical Modelling

Mathematical models are used to quantify governing laws of physical phenomena. These mathematical models are usually algebraic, ordinary, or partial differential equations. These equations consist of dependent and independent variables. Dependent variables are quantitive representations of physical properties derived from other physical properties. Let's consider a simple example of Newton's second law of motion, stating that "the rate of change of momentum of a body is equal to the net external force acting on it." Newton's second law of motion can be quantified as follows.

F=d(mv)/dt

This equation above, famously known as momentum conservation, is one of the fundamental equations in theoretical physics. The dependent variable in the above equation is the net external force acting on the body, and the independent variables are time, mass, and velocity of the body. Let us look at the different mathematical tools and their role in computational physics.

Linear Algebra and EigenValues

In computational physics, linear algebra is essential for solving and modeling physical systems. It involves efficient handling of systems of linear equations, matrix operations, and eigenvalue problems that arise in simulations and data analysis. Linear algebra tools such as matrix decomposition, iterative methods, and vector space analysis are crucial for solving equations related to fluid dynamics, quantum mechanics, and structural analysis. Physicists can accurately model complex systems, optimize computational algorithms, and interpret experimental data with greater precision utilizing the techniques of linear algebra. 

Ordinary Differential Equations

Ordinary differential equations (ODEs) play a crucial role in computational physics, as they often describe the dependence of physical systems on various physical parameters. From modeling the dynamics of particles under forces to simulating heat diffusion or fluid flow, ODEs provide a framework for understanding and predicting system behavior. Numerical methods, such as Euler's method, Runge-Kutta method, and adaptive step-size algorithms, are employed to solve these equations when analytical solutions are difficult to obtain. Accurate and efficient ODE solvers are essential for simulating complex physical phenomena and validating theoretical models in computational physics.

Partial Differential Equations

Partial differential equations (PDEs) are fundamental in computational physics, as they describe the physical phenomena involving multiple spatial dimensions and varying with time, such as heat conduction, fluid dynamics, and electromagnetic fields. PDEs are crucial for modeling complex systems where the behavior of variables depends on multiple variables. Numerical methods such as finite difference, finite element, and finite volume methods are used to obtain approximate solutions to PDEs at discrete points inside the domain, especially when analytical solutions to these equations are not feasible. These techniques enable physicists to simulate and analyze complex physical processes, making PDEs central to understanding and predicting a wide range of phenomena in computational physics.

Numerical Methods

Numerical methods are techniques for obtaining approximate solutions to different types of equations. The process of solving equations with numerical methods involves the discretization of governing equations and solving them at discrete points. The solution to these equations provides the numerical values of the independent variable at discrete points inside the domain. The ability of numerical methods to solve equations that can not be solved (certainly with available mathematical tools) analytically makes them quite helpful for predicting the behavior of physical systems defined by complex equations.

Programming in Computational Physics

Programming lies at the heart of computational physics, facilitating the implementation of mathematical models and algorithms into computational frameworks. Python, C++, and Fortran are the most commonly used programming languages due to their efficiency in numerical computation and the availability of robust libraries for scientific computing.

Through programming, physicists can build simulations that replicate real-world scenarios, adjust parameters, and analyze results with precision. The iterative nature of programming also allows for rapid prototyping and testing of new theories, providing a crucial bridge between theoretical insights and empirical observations. 

Applications Across Scientific Disciplines

Let us now take a look at the real-world applications of computational physics:

Astrophysics and Cosmology

In astrophysics and cosmology, computational simulation models aid in studying the formation and evolution of galaxies, the dynamics of star clusters, and the gravitational interactions between celestial bodies billions of light years away from our solar system. These simulations help astronomers interpret observational data from telescopes and satellites, Bringing clarity to the understanding of the origins and evolution of the universe.

Quantum Mechanics and Materials Science

Quantum mechanical simulations at the atomic and subatomic levels provide insights into the behavior of matter under extreme conditions. Computational methods help predict the properties of new materials, optimize chemical reactions, and design nanoscale devices with specific functionalities.

Climate Science and Environmental Modeling

Climate models use computational physics to simulate global weather patterns, ocean currents, and the impact of human activities on the Earth's climate system. These models inform policy decisions and Anticipate upcoming trends in temperature, precipitation, and sea level rise.

Biophysics and Medicine

In biophysics, computational modeling simulate biological processes such as protein folding, neuronal signaling, and the spread of infectious diseases. These models aid in drug discovery, personalized medicine, and understanding the mechanisms underlying complex biological systems.

Engineering and Fluid Dynamics

Engineering applications rely on computational fluid dynamics (CFD) to optimize aerodynamic designs, simulate heat transfer in mechanical systems, and predict the behavior of fluids in pipes and turbines. These simulations improve the efficiency and reliability of industrial processes and transportation systems.

References

1. https://www.newscientist.com/article/2160071-dancing-galaxies-may-shake-up-our-ideas-of-galaxy-formation/

2. https://www.scientificamerican.com/article/scientists-pick-the-most-beautiful-equations/

3. https://www.bairesdev.com/blog/choose-right-programming-language/

4. https://www.newscientist.com/definition/quantum-mechanics/

5. https://www.computationalfluiddynamics.com.au/tag/aerospace-and-defence/