In 1867, just two years after laying the foundations of electromagnetism, J. Clerk Maxwell presented a fundamental paper on kinetic gas theory, in which he described the evolution of the gas in terms of certain 'moments' of its velocity distribution function. This inspired Ludwig Boltzmann to formulate his famous kinetic equation, from which followed the H-theorem and the connection with entropy. On the occasion of the 150th anniversary of publication of Maxwell's paper, we review the Maxwell-Boltzmann formalism and discuss how its generality and adaptability enable it to play a key role in describing the behaviour of a variety of systems of current interest, in both gaseous and condensed matter, and in modern-day physics and technologies which Maxwell and Boltzmann could not possibly have foreseen. In particular, we illustrate the relevance and applicability of Maxwell's formalism to the dynamic field of plasma-wakefield acceleration.

Particle-beam-driven plasma wakefield acceleration (PWFA) enables various novel high-gradient techniques for powering future compact light-source and high-energy physics applications. Here, a driving particle bunch excites a wakefield response in a plasma medium, which may rapidly accelerate a trailing witness beam. In this Letter, we present the measurement of ratios of acceleration of the witness bunch to deceleration of the driver bunch, the so-called transformer ratio, significantly exceeding the fundamental theoretical and thus far experimental limit of 2 in a PWFA. An electron bunch with ramped current profile was utilized to accelerate a witness bunch with a transformer ratio of 4.6_{-0.7}^{+2.2} in a plasma with length ∼10  cm, also demonstrating stable transport of driver bunches with lengths on the order of the plasma wavelength.