Quantum dissipative effects for a real scalar field coupled to a dynamical Neumann surface in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> dimensions

We study dissipative effects for a system consisting of a massless real scalar field satisfying Neumann boundary conditions on a space and time-dependent surface, in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>d</a:mi> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:math> dimensions. We focus on the comparison of the results for this system with the ones corresponding to Dirichlet conditions, and the same surface spacetime geometry. We show that, in <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>d</c:mi> <c:mo>=</c:mo> <c:mn>1</c:mn> </c:math> , the effects are equal up to second order for rather arbitrary surfaces, and up to fourth order for wavelike surfaces. For <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>d</e:mi> <e:mo>&gt;</e:mo> <e:mn>1</e:mn> </e:math> , we find general expressions for their difference.