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Motivated by the challenge of sampling Gibbs measures with nonconvex
potentials, we study a continuum birth-death dynamics. We improve results in
previous works [51,57] and provide weaker hypotheses under which the
probability density of the birth-death governed by Kullback-Leibler divergence
or by $\chi^2$ divergence converge exponentially fast to the Gibbs equilibrium
measure, with a universal rate that is independent of the potential barrier. To
build a practical numerical sampler based on the pure birth-death dynamics, we
consider an interacting particle system, which is inspired by the gradient flow
structure and the classical Fokker-Planck equation and relies on kernel-based
approximations of the measure. Using the technique of $\Gamma$-convergence of
gradient flows, we show that on the torus, smooth and bounded positive
solutions of the kernelized dynamics converge on finite time intervals, to the
pure birth-death dynamics as the kernel bandwidth shrinks to zero. Moreover we
provide quantitative estimates on the bias of minimizers of the energy
corresponding to the kernelized dynamics. Finally we prove the long-time
asymptotic results on the convergence of the asymptotic states of the
kernelized dynamics towards the Gibbs measure.