Motivated by Voiculescu's liberation theory, we introduce the orbital free entropy χ orb for non-commutative self-adjoint random variables (also for "hyperfinite random multivariables"). Besides its basic properties, the relation of χ orb with the usual free entropy χ is shown. Moreover, the dimension counterpart δ 0,orb of χ orb is discussed, and we obtain the relation between δ 0,orb and the original free entropy dimension δ 0 together with applications to δ 0 itself. Throughout this section, let (M, τ) be a tracial W * -probability space and (X 1 , . . . , X n ) be an n-tuple of self-adjoint random variables in (M, τ). We will use the standard notations such as the microstate set Γ R (X 1 , . . . , X n ; N, m, δ) appearing in the course of defining the microstate free entropy χ(X 1 , . . . , X n ) (see [22]). We define a free entropy-like quantity as follows.Definition 2.1. For each δ > 0, m, N ∈ N, R > 0 and 1 ≤ i ≤ n, we denote byand by Γ orb,R (X 1 , . . . , X n ; N, m, δ) the set of all n-tuples (U 1 , . . . , U n ) of N × N unitary matrices such that there exists an n-tuple (D 1 , . . . , D n ) in n i=1 ∆ R (X i ; N, m, δ) satisfying
