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In this paper we extend to arbitrary number fields a construction of Bost-Connes of a -dynamical system with spontaneous symmetry breaking and partition function the Riemann zeta function.
    			                    
    			                 
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    			                    
                                       
In this article the linear Boltzmann equation is derived for a particle interacting with a Gaussian random field, in the weak coupling limit, with renewal in time of the random field. The initial data can be chosen arbitrarily. The proof is geometric and involves coherent states and semi-classical calculus.
    			                    
    			                 
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    
    		            
    		                
    		                
    		                
    			                
    		                
    		                
    		            
    			    			
    			 
 
    			
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