  We obtain some new results on the unimodal sequences of the real values of
rational functions by polynomials with positive integer coefficients. Thus, we
introduce the notion of merged-log-concavity of rational functions. Roughly
speaking, the notion extends Stanley's $q$-log-concavity of polynomials.
  We construct explicit merged-log-concave rational functions by $q$-binomial
coefficients, Hadamard products, and convolutions, extending the Cauchy-Binet
formula. Then, we obtain the unimodal sequences of rational functions by Young
diagrams. Moreover, we consider the variation of unimodal sequences by critical
points that separate strictly increasing, strictly decreasing, and hill-shape
sequences among almost strictly unimodal sequences. Also, the critical points
are zeros of polynomials in a suitable setting.
  The study above extends the $t$-power series of $(\pm t;q)_{\infty}^{\mp 1}$
to some extent by polynomials with positive integer coefficients and the
variation of unimodal sequences. We then obtain the golden ratio of quantum
dilogarithms ($q$-exponentials) as a critical point. Additionally, we consider
eta products, generalized Narayana numbers, and weighted $q$-multinomial
coefficients, which we introduce.
  In statistical mechanics, we discuss the grand canonical partition functions
of some ideal boson-fermion gases with or without Casimir energies (Ramanujan
summation). The merged-log-concavity gives phase transitions on Helmholtz free
energies by critical points of the metallic ratios including the golden ratio.
In particular, the phase transitions implies non-zero particle vacua from zero
particle vacua as the temperature rises.
