build.sbt

name := "valvar"

scalaVersion := "2.10.2" 

libraryDependencies ++= Seq( 
  "com.typesafe.akka" %% "akka-actor" % "2.2.3",  
  "com.typesafe.akka" %% "akka-testkit" % "2.2.3")

Main.scala

package valvar

import akka.actor.Actor
import akka.actor.Props
import collection.mutable.MutableList
import scala.concurrent.duration._
import scala.concurrent.ExecutionContext.Implicits.global

object Logger {
  case class Result(list: List[Int])
  case class ResultML(list: MutableList[Int])
  case class Log(list: List[Int])
  case class LogML(list: MutableList[Int])
}

class Logger extends Actor {
  import Logger._

  def receive: Receive = {
    case Result(list) => {
      context.system.scheduler.scheduleOnce(1000 millis, self, Log(list))
    }

    case ResultML(list) => {
      context.system.scheduler.scheduleOnce(1000 millis, self, LogML(list))
    }

    case Log(list) => println(list.mkString("Received var list: ", ", ", ""))

    case LogML(list) => println(list.mkString("Received mutable val list: ", ", ", ""))
  }
}

class Main extends Actor {
  import Logger._

  val logger = context.actorOf(Props[Logger], "logger")

  var varL = List(1, 2, 3)
  val valML = MutableList(1, 2, 3)

  println(varL.mkString("sent var list: ", ", ", ""))
  println(valML.mkString("sent mutable val list: ", ", ", ""))

  logger ! Result(varL)
  logger ! ResultML(valML)

  varL = varL :+ 23
  valML += 23

  println(varL.mkString("updated var list: ", ", ", ""))
  println(valML.mkString("updated mutable val list: ", ", ", ""))

  def receive = { case _ => }
}

output

sent var list: 1, 2, 3
sent mutable val list: 1, 2, 3
updated var list: 1, 2, 3, 23
updated mutable val list: 1, 2, 3, 23
Received var list: 1, 2, 3
Received mutable val list: 1, 2, 3, 23