Filippo Bovo


Statistical quantum systems and stochastic processes are my daily bread and butter. In particular, I focus on non-equilibrium and one-dimensional quantum systems. One can imagine an ink drop diffusing in a glass of water as an example of non-equilibrium system and a row of particles constrained to move on a line as an example of one-dimensional system.

Non-equilibrium systems


An ink drop falls in a glass of water under the action of gravity. Initially, the ink particles are just below the water surface and move downwards. As the ink falls down, some of the ink particles collide with the water particles and change their directions of motion: the number of ink particles moving downwards decreases and the number of particles moving in other directions increases. Eventually, if the ink has the same density of water, it diffuses into the water until it is homogeneously spread in an equilibrium state and the number of particles moving in every direction is the same. This example encompasses the three main features of the study of non-equilibrium quantum systems:

  • time evolution of the number of particles
  • diffusion
  • equilibration

Non-equilibrium classical and quantum systems are studied using the Keldysh technique, explained in goon fashion in the textbook ‘Field Theory of Non-Equilibrium Systems’ by A. Kamenev. You may find an alternative and simpler derivation of the formalism in my PhD thesis (link below).

One-dimensional systems


The beauty of one-dimensional systems relies on the following fact: particles moving on a line cannot avoid each other as would be possible in higher dimensions. As a consequence, any amount of repulsive interaction between particles leads to a sort of domino effect where a particle pushes the next one in line in a sequence that leads to a wave or other collective phenomena. However, unlike dominoes that, by falling, release energy when hit, particles in one dimension lose a little bit of energy between hits, leading to a wave that decays after some typically long time. Due to the main role of collisions in one dimension, the interaction between particles plays a fundamental role and the inescapable sequence of pushes leads to a strong correlation between different parts of the system. Another beautiful aspect of one-dimensional systems is that, unlike in higher dimensions, they are exactly solvable in several instances, lending themselves to a deeper comprehension.

A good introduction to equilibrium one-dimensional quantum systems is the textbook ‘Quantum Physics in One Dimension’ by T. Giamarchi. While this book uses the operator formalism, in my PhD thesis (link below) I give an alternative presentation using the Keldysh and functional integral formalisms. A good review of the dynamics of one-dimensional systems is ‘One-dimensional quantum liquids: Beyond the Luttinger liquid paradigm’ (Rev. Mod. Phys. 84, 1253) by A. Imambekov, T. Schmidt and L. Glazman.

Supervisor: Dr Dimitri Gangardt.

Keywords: Statistical physics, Stochastic Processes, Non-Equilibrium, Quantum physics, Condensed Matter Physics, Many-body quantum field theory, One-Dimensional Systems.

PhD Thesis: Thermal Dynamics of One-dimensional Quantum Systems

Publications: A full list can be found on the arXiv.


I correct undergraduate students' homework and give feedback on the following topics:

  • Fourier analysis
  • Complex analysis
  • Linear algebra
  • Differential equations
  • Integrals
  • Classical mechanics


Filippo Bovo

Research Student

+44 (0)121 414 4689
Room 403, Physics East